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# Functions that compute some of the sequences in Sloane’s tables

EXAMPLES:

Type sloane.[tab] to see a list of the sequences that are defined.

    sage: a = sloane.A000005; a
     The integer sequence tau(n), which is the number of divisors of n.
     sage: a(1)
     1
     sage: a(6)
     4
     sage: a(100)
     9

Type `d._eval??` to see how the function that computes an individual term of the sequence is implemented.

The input must be a positive integer:

    sage: a(0)
    Traceback (most recent call last):
    ...
    ValueError: input n (=0) must be a positive integer
    sage: a(1/3)
    Traceback (most recent call last):
    ...
    TypeError: input must be an int, long, or Integer

You can also change how a sequence prints:

    sage: a = sloane.A000005; a
    The integer sequence tau(n), which is the number of divisors of n.
    sage: a.rename('(..., tau(n), ...)')
    sage: a
    (..., tau(n), ...)
    sage: a.reset_name()
    sage: a
    The integer sequence tau(n), which is the number of divisors of n.

TESTS:

    sage: a = sloane.A000001
    sage: a == loads(dumps(a))
    True

    We agree with the online database::

    sage: for t in sloane.trait_names():    # long time; optional -- internet
    ....:     online_list = list(oeis(t).first_terms())
    ....:     L = max(2, len(online_list) // 2)
    ....:     sage_list = sloane.__getattribute__(t).list(L)
    ....:     if online_list[:L] != sage_list:
    ....:         print t, 'seems wrong'

See also

- If you want to get more informations relative to a sequence (references, links, examples, programs, ...), you can use the On-Line Encyclopedia of Integer Sequences provided by the `OEIS` module.
- If you plan to do a lot of automatic searches for subsequences, you should consider installing `SloaneEncyclopedia`, a local partial copy of the OEIS.

AUTHORS:

- William Stein: framework
- Jaap Spies: most sequences
- Nick Alexander: updated framework

class `sage.combinat.sloane_functions.``A000001`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Number of groups of order \(n\).

  Note: The database_gap-4.4.9 must be installed for \(n > 50\).

  run `sage -i database_gap-4.4.9` or higher first.

  INPUT:

  - `n` - positive integer

  OUTPUT: integer

  EXAMPLES:

      sage: a = sloane.A000001;a
      Number of groups of order n.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1) #optional database_gap
      1
      sage: a(2) #optional database_gap
      1
      sage: a(9) #optional database_gap
      2
      sage: a.list(16) #optional database_gap
      [1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14]
      sage: a(60)     # optional
      13

  AUTHORS:

  - Jaap Spies (2007-02-04)

class `sage.combinat.sloane_functions.``A000004`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  The zero sequence.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  EXAMPLES:

      sage: a = sloane.A000004; a
      The zero sequence.
      sage: a(1)
      0
      sage: a(2007)
      0
      sage: a.list(12)
      [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

  AUTHORS:

  - Jaap Spies (2006-12-10)

class `sage.combinat.sloane_functions.``A000005`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  The sequence \(tau(n)\), which is the number of divisors of \(n\).

  This sequence is also denoted \(d(n)\) (also called \(\tau(n)\) or \(\sigma_0(n)\)), the number of divisors of n.

  INPUT:

  - `n` - positive integer

  OUTPUT:

  EXAMPLES:

      sage: d = sloane.A000005; d
      The integer sequence tau(n), which is the number of divisors of n.
      sage: d(1)
      1
      sage: d(6)
      4
      sage: d(51)
      4
      sage: d(100)
      9
      sage: d(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: d.list(10)
      [1, 2, 2, 3, 2, 4, 2, 4, 3, 4]

  AUTHORS:

  - Jaap Spies (2006-12-10)
  - William Stein (2007-01-08)

class `sage.combinat.sloane_functions.``A000007`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  The characteristic function of 0: \(a(n) = 0^n\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000007;a
      The characteristic function of 0: a(n) = 0^n.
      sage: a(0)
      1
      sage: a(2)
      0
      sage: a(12)
      0
      sage: a.list(12)
      [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

  AUTHORS:

  - Jaap Spies (2007-01-12)

class `sage.combinat.sloane_functions.``A000008`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000008;a
      Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents.
      sage: a(0)
      1
      sage: a(1)
      1
      sage: a(13)
      16
      sage: a.list(14)
      [1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 11, 12, 15, 16]

  AUTHOR:

  - 10. Gaski (2009-05-29)

class `sage.combinat.sloane_functions.``A000009`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Number of partitions of \(n\) into odd parts.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000009;a
      Number of partitions of n into odd parts.
      sage: a(0)
      1
      sage: a(1)
      1
      sage: a(13)
      18
      sage: a.list(14)
      [1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]

  AUTHOR:

  - Jaap Spies (2007-01-30)
  `cf`()

    EXAMPLES:

        sage: it = sloane.A000009.cf()
        sage: [it.next() for i in range(14)]
        [1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]

  `list`(n)

    EXAMPLES:

        sage: sloane.A000009.list(14)
        [1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]

class `sage.combinat.sloane_functions.``A000010`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  The integer sequence A000010 is Euler’s totient function.

  Number of positive integers \(i < n\) that are relative prime to \(n\). Number of totatives of \(n\).

  Euler totient function \(\phi(n)\): count numbers \(n\) and prime to \(n\). euler_phi is a standard Sage function implemented in PARI

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000010; a
      Euler's totient function
      sage: a(1)
      1
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(11)
      10
      sage: a.list(12)
      [1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4]
      sage: a(1/3)
      Traceback (most recent call last):
      ...
      TypeError: input must be an int, long, or Integer

  AUTHORS:

  - Jaap Spies (2007-01-12)

class `sage.combinat.sloane_functions.``A000012`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  The all 1’s sequence.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000012; a
      The all 1's sequence.
      sage: a(1)
      1
      sage: a(2007)
      1
      sage: a.list(12)
      [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

  AUTHORS:

  - Jaap Spies (2007-01-12)

class `sage.combinat.sloane_functions.``A000015`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Smallest prime power \(\geq n\).

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000015; a
      Smallest prime power >= n.
      sage: a(1)
      1
      sage: a(8)
      8
      sage: a(305)
      307
      sage: a(-4)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-4) must be a positive integer
      sage: a.list(12)
      [1, 2, 3, 4, 5, 7, 7, 8, 9, 11, 11, 13]
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer

  AUTHORS:

  - Jaap Spies (2007-01-18)

class `sage.combinat.sloane_functions.``A000016`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Sloane’s A000016

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000016; a
      Sloane's A000016.
      sage: a(1)
      1
      sage: a(0)
      1
      sage: a(8)
      16
      sage: a(75)
      251859545753048193000
      sage: a(-4)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-4) must be an integer >= 0
      sage: a.list(12)
      [1, 1, 1, 2, 2, 4, 6, 10, 16, 30, 52, 94]

  AUTHORS:

  - Jaap Spies (2007-01-18)

class `sage.combinat.sloane_functions.``A000027`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  The natural numbers. Also called the whole numbers, the counting numbers or the positive integers.

  The following examples are tests of SloaneSequence more than A000027.

  EXAMPLES:

      sage: s = sloane.A000027; s
      The natural numbers.
      sage: s(10)
      10

  Index n is interpreted as _eval(n):

      sage: s[10]
      10

  Slices are interpreted with absolute offsets, so the following returns the terms of the sequence up to but not including the third term:

      sage: s[:3]
      [1, 2]
      sage: s[3:6]
      [3, 4, 5]
      sage: s.list(5)
      [1, 2, 3, 4, 5]

class `sage.combinat.sloane_functions.``A000030`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Initial digit of \(n\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000030; a
      Initial digit of n
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(8)
      8
      sage: a(454)
      4
      sage: a(-4)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-4) must be an integer >= 0
      sage: a.list(12)
      [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1]

  AUTHORS:

  - Jaap Spies (2007-01-18)

class `sage.combinat.sloane_functions.``A000032`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Lucas numbers (beginning at 2): \(L(n) = L(n-1) + L(n-2)\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000032; a
      Lucas numbers (beginning at 2): L(n) = L(n-1) + L(n-2).
      sage: a(0)
      2
      sage: a(1)
      1
      sage: a(8)
      47
      sage: a(200)
      627376215338105766356982006981782561278127
      sage: a(-4)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-4) must be an integer >= 0
      sage: a.list(12)
      [2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199]

  AUTHORS:

  - Jaap Spies (2007-01-18)

class `sage.combinat.sloane_functions.``A000035`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  A simple periodic sequence.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000035;a
      A simple periodic sequence.
      sage: a(0.0)
      Traceback (most recent call last):
      ...
      TypeError: input must be an int, long, or Integer
      sage: a(1)
      1
      sage: a(2)
      0
      sage: a(9)
      1
      sage: a.list(10)
      [0, 1, 0, 1, 0, 1, 0, 1, 0, 1]

  AUTHORS:

  - Jaap Spies (2007-02-02)

class `sage.combinat.sloane_functions.``A000040`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  The prime numbers.

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000040; a
      The prime numbers.
      sage: a(1)
      2
      sage: a(8)
      19
      sage: a(305)
      2011
      sage: a.list(12)
      [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37]
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer

  AUTHORS:

  - Jaap Spies (2007-01-17)

class `sage.combinat.sloane_functions.``A000041`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  \(a(n)\) = number of partitions of \(n\) (the partition numbers).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000041;a
      a(n) = number of partitions of n (the partition numbers).
      sage: a(0)
      1
      sage: a(2)
      2
      sage: a(8)
      22
      sage: a(200)
      3972999029388
      sage: a.list(9)
      [1, 1, 2, 3, 5, 7, 11, 15, 22]

  AUTHORS:

  - Jaap Spies (2007-01-18)

class `sage.combinat.sloane_functions.``A000043`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Primes \(p\) such that \(2^p - 1\) is prime. \(2^p - 1\) is then called a Mersenne prime.

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000043;a
      Primes p such that 2^p - 1 is prime. 2^p - 1 is then called a Mersenne prime.
      sage: a(1)
      2
      sage: a(2)
      3
      sage: a(39)
      13466917
      sage: a(40)
      Traceback (most recent call last):
      ...
      IndexError: list index out of range
      sage: a.list(12)
      [2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class `sage.combinat.sloane_functions.``A000045`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Sequence of Fibonacci numbers, offset 0,4.

  REFERENCES:

  - S. Plouffe, Project Gutenberg, The First 1001 Fibonacci Numbers, http://ibiblio.org/pub/docs/books/gutenberg/etext01/fbncc10.txt

  We have one more. Our first Fibonacci number is 0.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000045; a
      Fibonacci numbers with index n >= 0
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a.list(12)
      [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89]
      sage: a(1/3)
      Traceback (most recent call last):
      ...
      TypeError: input must be an int, long, or Integer

  AUTHORS:

  - Jaap Spies (2007-01-13)
  `fib`()

    Returns a generator over all Fibonacci numbers, starting with 0.

    EXAMPLES:

        sage: it = sloane.A000045.fib()
        sage: [it.next() for i in range(10)]
        [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]

  `list`(n)

    EXAMPLES:

        sage: sloane.A000045.list(10)
        [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]

class `sage.combinat.sloane_functions.``A000069`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Odious numbers: odd number of 1’s in binary expansion.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000069; a
      Odious numbers: odd number of 1's in binary expansion.
      sage: a(0)
      1
      sage: a(2)
      4
      sage: a.list(9)
      [1, 2, 4, 7, 8, 11, 13, 14, 16]

  AUTHORS:

  - Jaap Spies (2007-02-02)

class `sage.combinat.sloane_functions.``A000073`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3). Starting with 0, 0, 1, ...

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000073;a
      Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3).
      sage: a(0)
      0
      sage: a(1)
      0
      sage: a(2)
      1
      sage: a(11)
      149
      sage: a.list(12)
      [0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149]

  AUTHORS:

  - Jaap Spies (2007-01-19)
  `list`(n)

    EXAMPLES:

        sage: sloane.A000073.list(10)
        [0, 0, 1, 1, 2, 4, 7, 13, 24, 44]

class `sage.combinat.sloane_functions.``A000079`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Powers of 2: \(a(n) = 2^n\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000079;a
      Powers of 2: a(n) = 2^n.
      sage: a(0)
      1
      sage: a(2)
      4
      sage: a(8)
      256
      sage: a(100)
      1267650600228229401496703205376
      sage: a.list(9)
      [1, 2, 4, 8, 16, 32, 64, 128, 256]

  AUTHORS:

  - Jaap Spies (2007-01-18)

class `sage.combinat.sloane_functions.``A000085`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Number of self-inverse permutations on \(n\) letters, also known as involutions; number of Young tableaux with \(n\) cells.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000085;a
      Number of self-inverse permutations on n letters.
      sage: a(0)
      1
      sage: a(1)
      1
      sage: a(2)
      2
      sage: a(12)
      140152
      sage: a.list(13)
      [1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35696, 140152]

  AUTHORS:

  - Jaap Spies (2007-02-03)

class `sage.combinat.sloane_functions.``A000100`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000100;a
      Number of compositions of n in which the maximum part size is 3.
      sage: a(0)
      0
      sage: a(1)
      0
      sage: a(2)
      0
      sage: a(3)
      1
      sage: a(11)
      360
      sage: a.list(12)
      [0, 0, 0, 1, 2, 5, 11, 23, 47, 94, 185, 360]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class `sage.combinat.sloane_functions.``A000108`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Catalan numbers: \(C_n = \frac{{{2n}\choose{n}}}{n+1} = \frac {(2n)!}{n!(n+1)!}\). Also called Segner numbers.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000108;a
      Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Also called Segner numbers.
      sage: a(0)
      1
      sage: a.offset
      0
      sage: a(8)
      1430
      sage: a(40)
      2622127042276492108820
      sage: a.list(9)
      [1, 1, 2, 5, 14, 42, 132, 429, 1430]

  AUTHORS:

  - Jaap Spies (2007-01-12)

class `sage.combinat.sloane_functions.``A000110`

  Bases: `sage.combinat.sloane_functions.ExponentialNumbers`

  The sequence of Bell numbers.

  The Bell number \(B_n\) counts the number of ways to put \(n\) distinguishable things into indistinguishable boxes such that no box is empty.

  Let \(S(n, k)\) denote the Stirling number of the second kind. Then

  \[B_n = \sum{k=0}^{n} S(n, k) .\]

  INPUT:

  - `n` - integer = 0

  OUTPUT:

  - `integer` - \(B_n\)

  EXAMPLES:

      sage: a = sloane.A000110; a
      Sequence of Bell numbers
      sage: a.offset
      0
      sage: a(0)
      1
      sage: a(100)
      47585391276764833658790768841387207826363669686825611466616334637559114497892442622672724044217756306953557882560751
      sage: a.list(10)
      [1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147]

  AUTHORS:

  - Nick Alexander

class `sage.combinat.sloane_functions.``A000120`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  1’s-counting sequence: number of 1’s in binary expansion of \(n\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000120;a
      1's-counting sequence: number of 1's in binary expansion of n.
      sage: a(0)
      0
      sage: a(2)
      1
      sage: a(12)
      2
      sage: a.list(12)
      [0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3]

  AUTHORS:

  - Jaap Spies (2007-01-26)
  `f`(n)

    EXAMPLES:

        sage: [sloane.A000120.f(n) for n in range(10)]
        [0, 1, 1, 2, 1, 2, 2, 3, 1, 2]

class `sage.combinat.sloane_functions.``A000124`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Central polygonal numbers (the Lazy Caterer’s sequence): \(n(n+1)/2 + 1\).

  Or, maximal number of pieces formed when slicing a pancake with \(n\) cuts.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000124;a
      Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1.
      sage: a(0)
      1
      sage: a(1)
      2
      sage: a(2)
      4
      sage: a(9)
      46
      sage: a.list(10)
      [1, 2, 4, 7, 11, 16, 22, 29, 37, 46]

  AUTHORS:

  - Jaap Spies (2007-01-25)

class `sage.combinat.sloane_functions.``A000129`

  Bases: `sage.combinat.sloane_functions.RecurrenceSequence2`

  Pell numbers: \(a(0) = 0\), \(a(1) = 1\); for \(n > 1\), \(a(n) = 2a(n-1) + a(n-2)\).

  Denominators of continued fraction convergents to \(\sqrt 2\).

  See also A001333

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000129;a
      Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).
      sage: a(0)
      0
      sage: a(2)
      2
      sage: a(12)
      13860
      sage: a.list(12)
      [0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741]

  AUTHORS:

  - Jaap Spies (2007-01-25)

class `sage.combinat.sloane_functions.``A000142`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Factorial numbers: \(n! = 1 \cdot 2 \cdot 3 \cdots n\)

  Order of symmetric group \(S_n\), number of permutations of \(n\) letters.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000142;a
      Factorial numbers: n! = 1*2*3*4*...*n (order of symmetric group S_n, number of permutations of n letters).
      sage: a(0)
      1
      sage: a(8)
      40320
      sage: a(40)
      815915283247897734345611269596115894272000000000
      sage: a.list(9)
      [1, 1, 2, 6, 24, 120, 720, 5040, 40320]

  AUTHORS:

  - Jaap Spies (2007-01-12)

class `sage.combinat.sloane_functions.``A000153`

  Bases: `sage.combinat.sloane_functions.ExtremesOfPermanentsSequence`

  \(a(n) = n*a(n-1) + (n-2)*a(n-2)\), with \(a(0) = 0\), \(a(1) = 1\).

  With offset 1, permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=2\) and \(n\) zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000153; a
      a(n) = n*a(n-1) + (n-2)*a(n-2), with a(0) = 0, a(1) = 1.
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(8)
      82508
      sage: a(20)
      10315043624498196944
      sage: a.list(8)
      [0, 1, 2, 7, 32, 181, 1214, 9403]

  AUTHORS:

  - Jaap Spies (2007-01-13)

class `sage.combinat.sloane_functions.``A000165`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Double factorial numbers: \((2n)!! = 2^n*n!\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000165;a
      Double factorial numbers: (2n)!! = 2^n*n!.
      sage: a(0)
      1
      sage: a.offset
      0
      sage: a(8)
      10321920
      sage: a(20)
      2551082656125828464640000
      sage: a.list(9)
      [1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920]

  AUTHORS:

  - Jaap Spies (2007-01-24)

class `sage.combinat.sloane_functions.``A000166`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Subfactorial or rencontres numbers, or derangements: number of permutations of \(n\) elements with no fixed points.

  With offset 1 also the permanent of a (0,1)-matrix of order \(n\) with \(n\) 0’s not on a line.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000166;a
      Subfactorial or rencontres numbers, or derangements: number of permutations of $n$ elements with no fixed points.
      sage: a(0)
      1
      sage: a(1)
      0
      sage: a(2)
      1
      sage: a.offset
      0
      sage: a(8)
      14833
      sage: a(20)
      895014631192902121
      sage: a.list(9)
      [1, 0, 1, 2, 9, 44, 265, 1854, 14833]

  AUTHORS:

  - Jaap Spies (2007-01-13)

class `sage.combinat.sloane_functions.``A000169`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Number of labeled rooted trees with \(n\) nodes: \(n^{(n-1)}\).

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000169;a
      Number of labeled rooted trees with n nodes: n^(n-1).
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      1
      sage: a(2)
      2
      sage: a(10)
      1000000000
      sage: a.list(11)
      [1, 2, 9, 64, 625, 7776, 117649, 2097152, 43046721, 1000000000, 25937424601]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class `sage.combinat.sloane_functions.``A000203`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  The sequence \(\sigma(n)\), where \(\sigma(n)\) is the sum of the divisors of \(n\). Also called \(\sigma_1(n)\).

  The function `sigma(n, k)` implements \(\sigma_k(n)\) in Sage.

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000203; a
      sigma(n) = sum of divisors of n. Also called sigma_1(n).
      sage: a(1)
      1
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(256)
      511
      sage: a.list(12)
      [1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28]
      sage: a(1/3)
      Traceback (most recent call last):
      ...
      TypeError: input must be an int, long, or Integer

  AUTHORS:

  - Jaap Spies (2007-01-13)

class `sage.combinat.sloane_functions.``A000204`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Lucas numbers (beginning with 1): \(L(n) = L(n-1) + L(n-2)\) with \(L(1) = 1\), \(L(2) = 3\).

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000204; a
      Lucas numbers (beginning at 1): L(n) = L(n-1) + L(n-2), L(2) = 3.
      sage: a(1)
      1
      sage: a(8)
      47
      sage: a(200)
      627376215338105766356982006981782561278127
      sage: a(-4)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-4) must be a positive integer
      sage: a.list(12)
      [1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322]
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer

  AUTHORS:

  - Jaap Spies (2007-01-18)

class `sage.combinat.sloane_functions.``A000213`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3). Starting with 1, 1, 1, ...

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000213;a
      Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3).
      sage: a(0)
      1
      sage: a(1)
      1
      sage: a(2)
      1
      sage: a(11)
      355
      sage: a.list(12)
      [1, 1, 1, 3, 5, 9, 17, 31, 57, 105, 193, 355]

  AUTHORS:

  - Jaap Spies (2007-01-19)
  `list`(n)

    EXAMPLES:

        sage: sloane.A000213.list(10)
        [1, 1, 1, 3, 5, 9, 17, 31, 57, 105]

class `sage.combinat.sloane_functions.``A000217`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Triangular numbers: \(a(n) = {n+1} \choose 2) = n(n+1)/2\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000217;a
      Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n.
      sage: a(0)
      0
      sage: a(2)
      3
      sage: a(8)
      36
      sage: a(2000)
      2001000
      sage: a.list(9)
      [0, 1, 3, 6, 10, 15, 21, 28, 36]

  AUTHORS:

  - Jaap Spies (2007-01-25)

class `sage.combinat.sloane_functions.``A000225`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  \(2^n - 1\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000225;a
      2^n - 1.
      sage: a(0)
      0
      sage: a(-1)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be an integer >= 0
      sage: a(12)
      4095
      sage: a.list(12)
      [0, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047]

  AUTHORS:

  - Jaap Spies (2007-01-25)

class `sage.combinat.sloane_functions.``A000244`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Powers of 3: \(a(n) = 3^n\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000244;a
      Powers of 3: a(n) = 3^n.
      sage: a(-1)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be an integer >= 0
      sage: a(0)
      1
      sage: a(3)
      27
      sage: a(11)
      177147
      sage: a.list(12)
      [1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class `sage.combinat.sloane_functions.``A000255`

  Bases: `sage.combinat.sloane_functions.ExtremesOfPermanentsSequence`

  \(a(n) = n*a(n-1) + (n-1)*a(n-2)\), with \(a(0) = 1\), \(a(1) = 1\).

  With offset 1, permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=1\) and \(n\) zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000255;a
      a(n) = n*a(n-1) + (n-1)*a(n-2), a(0) = 1, a(1) = 1.
      sage: a(0)
      1
      sage: a(1)
      1
      sage: a.offset
      0
      sage: a(8)
      148329
      sage: a(22)
      9923922230666898717143
      sage: a.list(9)
      [1, 1, 3, 11, 53, 309, 2119, 16687, 148329]

  AUTHORS:

  - Jaap Spies (2007-01-13)

class `sage.combinat.sloane_functions.``A000261`

  Bases: `sage.combinat.sloane_functions.ExtremesOfPermanentsSequence`

  \(a(n) = n*a(n-1) + (n-3)*a(n-2)\), with \(a(1) = 1\), \(a(2) = 1\).

  With offset 1, permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=3\) and \(n\) zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

  Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000261;a
      a(n) = n*a(n-1) + (n-3)*a(n-2), a(1) = 0, a(2) = 1.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      0
      sage: a.offset
      1
      sage: a(8)
      30637
      sage: a(22)
      1801366114380914335441
      sage: a.list(9)
      [0, 1, 3, 13, 71, 465, 3539, 30637, 296967]

  AUTHORS:

  - Jaap Spies (2007-01-23)

class `sage.combinat.sloane_functions.``A000272`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Number of labeled rooted trees on \(n\) nodes: \(n^{(n-2)}\).

  INPUT:

  - `n` - integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000272;a
      Number of labeled rooted trees with n nodes: n^(n-2).
      sage: a(0)
      1
      sage: a(1)
      1
      sage: a(2)
      1
      sage: a(10)
      100000000
      sage: a.list(12)
      [1, 1, 1, 3, 16, 125, 1296, 16807, 262144, 4782969, 100000000, 2357947691]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class `sage.combinat.sloane_functions.``A000290`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  The squares: \(a(n) = n^2\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000290;a
      The squares: a(n) = n^2.
      sage: a(0)
      0
      sage: a(-1)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be an integer >= 0
      sage: a(16)
      256
      sage: a.list(17)
      [0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256]

  AUTHORS:

  - Jaap Spies (2007-01-25)

class `sage.combinat.sloane_functions.``A000292`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Tetrahedral (or pyramidal) numbers: \({n+2} \choose 3 = n(n+1)(n+2)/6\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000292;a
      Tetrahedral (or pyramidal) numbers: C(n+2,3) = n(n+1)(n+2)/6.
      sage: a(0)
      0
      sage: a(2)
      4
      sage: a(11)
      286
      sage: a.list(12)
      [0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class `sage.combinat.sloane_functions.``A000302`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Powers of 4: \(a(n) = 4^n\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000302;a
      Powers of 4: a(n) = 4^n.
      sage: a(0)
      1
      sage: a(1)
      4
      sage: a(2)
      16
      sage: a(10)
      1048576
      sage: a.list(12)
      [1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1048576, 4194304]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class `sage.combinat.sloane_functions.``A000312`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Number of labeled mappings from \(n\) points to themselves (endofunctions): \(n^n\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000312;a
      Number of labeled mappings from n points to themselves (endofunctions): n^n.
      sage: a(-1)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be an integer >= 0
      sage: a(0)
      1
      sage: a(1)
      1
      sage: a(9)
      387420489
      sage: a.list(11)
      [1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 10000000000]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class `sage.combinat.sloane_functions.``A000326`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Pentagonal numbers: \(n(3n-1)/2\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000326;a
      Pentagonal numbers: n(3n-1)/2.
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(2)
      5
      sage: a(10)
      145
      sage: a.list(12)
      [0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176]
      sage: a(1/3)
      Traceback (most recent call last):
      ...
      TypeError: input must be an int, long, or Integer

  AUTHORS:

  - Jaap Spies (2007-01-26)

class `sage.combinat.sloane_functions.``A000330`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Square pyramidal numbers” \(0^2 + 1^2 \cdots n^2 = n(n+1)(2n+1)/6\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000330;a
      Square pyramidal numbers: 0^2+1^2+2^2+...+n^2 = n(n+1)(2n+1)/6.
      sage: a(-1)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be an integer >= 0
      sage: a(0)
      0
      sage: a(3)
      14
      sage: a(11)
      506
      sage: a.list(12)
      [0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class `sage.combinat.sloane_functions.``A000396`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Perfect numbers: equal to sum of proper divisors.

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000396;a
      Perfect numbers: equal to sum of proper divisors.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      6
      sage: a(2)
      28
      sage: a(7)
      137438691328
      sage: a.list(7)
      [6, 28, 496, 8128, 33550336, 8589869056, 137438691328]

  AUTHORS:

  - Jaap Spies (2007-01-25)

class `sage.combinat.sloane_functions.``A000578`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  The cubes: \(a(n) = n^3\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000578;a
      The cubes: n^3
      sage: a(-1)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be an integer >= 0
      sage: a(0)
      0
      sage: a(3)
      27
      sage: a(11)
      1331
      sage: a.list(12)
      [0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class `sage.combinat.sloane_functions.``A000583`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Fourth powers: \(a(n) = n^4\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000583;a
      Fourth powers: n^4.
      sage: a(0.0)
      Traceback (most recent call last):
      ...
      TypeError: input must be an int, long, or Integer
      sage: a(1)
      1
      sage: a(2)
      16
      sage: a(9)
      6561
      sage: a.list(10)
      [0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561]

  AUTHORS:

  - Jaap Spies (2007-02-04)

class `sage.combinat.sloane_functions.``A000587`

  Bases: `sage.combinat.sloane_functions.ExponentialNumbers`

  The sequence of Uppuluri-Carpenter numbers.

  The Uppuluri-Carpenter number \(C_n\) counts the imbalance in the number of ways to put \(n\) distinguishable things into an even number of indistinguishable boxes versus into an odd number of indistinguishable boxes, such that no box is empty.

  Let \(S(n, k)\) denote the Stirling number of the second kind. Then

  \[C_n = \sum{k=0}^{n} (-1)^k S(n, k) .\]

  INPUT:

  - `n` - integer = 0

  OUTPUT:

  - `integer` - \(C_n\)

  EXAMPLES:

      sage: a = sloane.A000587; a
      Sequence of Uppuluri-Carpenter numbers
      sage: a.offset
      0
      sage: a(0)
      1
      sage: a(100)
      397577026456518507969762382254187048845620355238545130875069912944235105204434466095862371032124545552161
      sage: a.list(10)
      [1, -1, 0, 1, 1, -2, -9, -9, 50, 267]

  AUTHORS:

  - Nick Alexander

class `sage.combinat.sloane_functions.``A000668`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Mersenne primes (of form \(2^p - 1\) where \(p\) is a prime).

  (See A000043 for the values of \(p\).)

  Warning: a(39) has 4,053,946 digits!

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000668;a
      Mersenne primes (of form 2^p - 1 where p is a prime). (See A000043 for the values of p.)
      sage: a(1)
      3
      sage: a(2)
      7
      sage: a(12)
      170141183460469231731687303715884105727

  Warning: a(39) has 4,053,946 digits!

      sage: a(40)
      Traceback (most recent call last):
      ...
      IndexError: list index out of range
      sage: a.list(8)
      [3, 7, 31, 127, 8191, 131071, 524287, 2147483647]

  AUTHORS:

  - Jaap Spies (2007-01-25)

class `sage.combinat.sloane_functions.``A000670`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Number of preferential arrangements of \(n\) labeled elements; or number of weak orders on \(n\) labeled elements.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000670;a
      Number of preferential arrangements of n labeled elements.
      sage: a(0)
      1
      sage: a(1)
      1
      sage: a(2)
      3
      sage: a(9)
      7087261
      sage: a.list(10)
      [1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261]

  AUTHORS:

  - Jaap Spies (2007-02-03)

class `sage.combinat.sloane_functions.``A000720`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  \(pi(n)\), the number of primes \(\le n\). Sometimes called \(PrimePi(n)\).

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000720;a
      pi(n), the number of primes <= n. Sometimes called PrimePi(n)
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(2)
      1
      sage: a(8)
      4
      sage: a(1000)
      168
      sage: a.list(12)
      [0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5]

  AUTHORS:

  - Jaap Spies (2007-01-25)

class `sage.combinat.sloane_functions.``A000796`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Decimal expansion of \(\pi\).

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000796;a
      Decimal expansion of Pi.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      3
      sage: a(13)
      9
      sage: a.list(14)
      [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7]
      sage: a(100)
      7

  AUTHOR:

  - Jaap Spies (2007-01-30)
  `list`(n)

    EXAMPLES:

        sage: sloane.A000796.list(10)
        [3, 1, 4, 1, 5, 9, 2, 6, 5, 3]

  `pi`()

    Based on an algorithm of Lambert Meertens The ABC-programming language!!!

    EXAMPLES:

        sage: it = sloane.A000796.pi()
        sage: [it.next() for i in range(10)]
        [3, 1, 4, 1, 5, 9, 2, 6, 5, 3]

class `sage.combinat.sloane_functions.``A000961`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Prime powers

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000961;a
      Prime powers.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(2)
      2
      sage: a(12)
      17
      sage: a.list(12)
      [1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17]

  AUTHORS:

  - Jaap Spies (2007-01-25)
  `list`(n)

    EXAMPLES:

        sage: sloane.A000961.list(10)
        [1, 2, 3, 4, 5, 7, 8, 9, 11, 13]

class `sage.combinat.sloane_functions.``A000984`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Central binomial coefficients: \(2n \choose n = \frac {(2n)!} {(n!)^2}\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A000984;a
      Central binomial coefficients: C(2n,n) = (2n)!/(n!)^2
      sage: a(0)
      1
      sage: a(2)
      6
      sage: a(8)
      12870
      sage: a.list(9)
      [1, 2, 6, 20, 70, 252, 924, 3432, 12870]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class `sage.combinat.sloane_functions.``A001006`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Motzkin numbers: number of ways of drawing any number of nonintersecting chords among \(n\) points on a circle.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A001006;a
      Motzkin numbers: number of ways of drawing any number of nonintersecting chords among n points on a circle.
      sage: a(0)
      1
      sage: a(1)
      1
      sage: a(2)
      2
      sage: a(12)
      15511
      sage: a.list(13)
      [1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511]

  AUTHORS:

  - Jaap Spies (2007-02-02)

class `sage.combinat.sloane_functions.``A001045`

  Bases: `sage.combinat.sloane_functions.RecurrenceSequence2`

  Jacobsthal sequence: \(a(n) = a(n-1) + 2a(n-2)\), \(a(0) = 0\) and \(a(1) = 1\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A001045;a
      Jacobsthal sequence: a(n) = a(n-1) + 2a(n-2).
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(2)
      1
      sage: a(11)
      683
      sage: a.list(12)
      [0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class `sage.combinat.sloane_functions.``A001055`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Number of ways of factoring \(n\) with all factors 1.

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A001055;a
      Number of ways of factoring n with all factors >1.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      1
      sage: a(2)
      1
      sage: a(9)
      2
      sage: a.list(16)
      [1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5]

  AUTHORS:

  - Jaap Spies (2007-02-04)
  `nwf`(n, m)

    EXAMPLES:

        sage: sloane.A001055.nwf(4,1)
        0
        sage: sloane.A001055.nwf(4,2)
        1
        sage: sloane.A001055.nwf(4,3)
        1
        sage: sloane.A001055.nwf(4,4)
        2

class `sage.combinat.sloane_functions.``A001109`

  Bases: `sage.combinat.sloane_functions.RecurrenceSequence2`

  \(a(n)^2\) is a triangular number: \(a(n) = 6*a(n-1) - a(n-2)\) with \(a(0)=0\), \(a(1)=1\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A001109;a
      a(n)^2 is a triangular number: a(n) = 6*a(n-1) - a(n-2) with a(0)=0, a(1)=1
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(2)
      6
      sage: a.offset
      0
      sage: a(8)
      235416
      sage: a(60)
      1515330104844857898115857393785728383101709300
      sage: a.list(9)
      [0, 1, 6, 35, 204, 1189, 6930, 40391, 235416]

  AUTHORS:

  - Jaap Spies (2007-01-24)

class `sage.combinat.sloane_functions.``A001110`

  Bases: `sage.combinat.sloane_functions.RecurrenceSequence`

  Numbers that are both triangular and square: \(a(n) = 34a(n-1) - a(n-2) + 2\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A001110; a
      Numbers that are both triangular and square: a(n) = 34a(n-1) - a(n-2) + 2.
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(8)
      55420693056
      sage: a(21)
      4446390382511295358038307980025
      sage: a.list(8)
      [0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881]

  AUTHORS:

  - Jaap Spies (2007-01-19)
  `g`(k)

    EXAMPLES:

        sage: sloane.A001110.g(2)
        2
        sage: sloane.A001110.g(1)
        0

class `sage.combinat.sloane_functions.``A001147`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Double factorial numbers: \((2n-1)!! = 1 \cdot 3 \cdot 5 \cdots (2n-1)\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A001147;a
      Double factorial numbers: (2n-1)!! = 1.3.5....(2n-1).
      sage: a(0)
      1
      sage: a.offset
      0
      sage: a(8)
      2027025
      sage: a(20)
      319830986772877770815625
      sage: a.list(9)
      [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025]

  AUTHORS:

  - Jaap Spies (2007-01-24)

class `sage.combinat.sloane_functions.``A001157`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  The sequence \(\sigma_2(n)\), sum of squares of divisors of \(n\).

  The function sigma(n, k) implements \(\sigma_k*\) in Sage.

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A001157;a
      sigma_2(n): sum of squares of divisors of n
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(2)
      5
      sage: a(8)
      85
      sage: a.list(9)
      [1, 5, 10, 21, 26, 50, 50, 85, 91]

  AUTHORS:

  - Jaap Spies (2007-01-13)

class `sage.combinat.sloane_functions.``A001189`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Number of degree-n permutations of order exactly 2.

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A001189;a
      Number of degree-n permutations of order exactly 2.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      0
      sage: a(2)
      1
      sage: a(12)
      140151
      sage: a.list(13)
      [0, 1, 3, 9, 25, 75, 231, 763, 2619, 9495, 35695, 140151, 568503]

  AUTHORS:

  - Jaap Spies (2007-02-03)

class `sage.combinat.sloane_functions.``A001221`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Number of different prime divisors of \(n\)

  Also called omega(n) or \(\omega(n)\). Maximal number of terms in any factorization of \(n\). Number of prime powers that divide \(n\).

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A001221; a
      Number of distinct primes dividing n (also called omega(n)).
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      0
      sage: a(8)
      1
      sage: a(41)
      1
      sage: a(84792)
      3
      sage: a.list(12)
      [0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2]

  AUTHORS:

  - Jaap Spies (2007-01-19)

class `sage.combinat.sloane_functions.``A001222`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Number of prime divisors of \(n\) (counted with multiplicity).

  Also called bigomega(n) or \(\Omega(n)\). Maximal number of terms in any factorization of \(n\). Number of prime powers that divide \(n\).

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A001222; a
      Number of prime divisors of n (counted with multiplicity).
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      0
      sage: a(8)
      3
      sage: a(41)
      1
      sage: a(84792)
      5
      sage: a.list(12)
      [0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3]

  AUTHORS:

  - Jaap Spies (2007-01-19)

class `sage.combinat.sloane_functions.``A001227`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Number of odd divisors of \(n\).

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A001227; a
      Number of odd divisors of n
      sage: a.offset
      1
      sage: a(1)
      1
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(100)
      3
      sage: a(256)
      1
      sage: a(29)
      2
      sage: a.list(20)
      [1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 4, 1, 2, 3, 2, 2]
      sage: a(-1)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be a positive integer

  AUTHORS:

  - Jaap Spies (2007-01-14)

class `sage.combinat.sloane_functions.``A001333`

  Bases: `sage.combinat.sloane_functions.RecurrenceSequence2`

  Numerators of continued fraction convergents to \(\sqrt 2\).

  See also A000129

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A001333;a
      Numerators of continued fraction convergents to sqrt(2).
      sage: a(0)
      1
      sage: a(1)
      1
      sage: a(2)
      3
      sage: a(3)
      7
      sage: a(11)
      8119
      sage: a.list(12)
      [1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, 8119]

  AUTHORS:

  - Jaap Spies (2007-02-01)

class `sage.combinat.sloane_functions.``A001358`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Products of two primes.

  These numbers have been called semiprimes (or semi-primes), biprimes or 2-almost primes.

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A001358;a
      Products of two primes.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(2)
      6
      sage: a(8)
      22
      sage: a(200)
      669
      sage: a.list(9)
      [4, 6, 9, 10, 14, 15, 21, 22, 25]

  AUTHORS:

  - Jaap Spies (2007-01-25)
  `list`(n)

    EXAMPLES:

        sage: sloane.A001358.list(9)
        [4, 6, 9, 10, 14, 15, 21, 22, 25]

class `sage.combinat.sloane_functions.``A001405`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Central binomial coefficients: \(n \choose \lfloor \frac {n}{ 2} \rfloor\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A001405;a
      Central binomial coefficients: C(n,floor(n/2)).
      sage: a(0)
      1
      sage: a(2)
      2
      sage: a(12)
      924
      sage: a.list(12)
      [1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, 462]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class `sage.combinat.sloane_functions.``A001477`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  The nonnegative integers.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A001477;a
      The nonnegative integers.
      sage: a(-1)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be an integer >= 0
      sage: a(0)
      0
      sage: a(3382789)
      3382789
      sage: a(11)
      11
      sage: a.list(12)
      [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class `sage.combinat.sloane_functions.``A001694`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  This function returns the \(n\)-th Powerful Number:

  A positive integer \(n\) is powerful if for every prime \(p\) dividing \(n\), \(p^2\) also divides \(n\).

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A001694; a
      Powerful Numbers (also called squarefull, square-full or 2-full numbers).
      sage: a.offset
      1
      sage: a(1)
      1
      sage: a(4)
      9
      sage: a(100)
      3136
      sage: a(156)
      7225
      sage: a.list(19)
      [1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144]
      sage: a(-1)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be a positive integer

  AUTHORS:

  - Jaap Spies (2007-01-14)
  `is_powerful`(n)

    This function returns True if and only if \(n\) is a Powerful Number:

    A positive integer \(n\) is powerful if for every prime \(p\) dividing \(n\), \(p^2\) also divides \(n\). See Sloane’s OEIS A001694.

    INPUT:

    - `n` - integer

    OUTPUT:

    - `True` - if \(n\) is a Powerful number, else False

    EXAMPLES:

        sage: a = sloane.A001694
        sage: a.is_powerful(2500)
        True
        sage: a.is_powerful(20)
        False

    AUTHORS:

    - Jaap Spies (2006-12-07)
  `list`(n)

    EXAMPLES:

        sage: sloane.A001694.list(9)
        [1, 4, 8, 9, 16, 25, 27, 32, 36]

class `sage.combinat.sloane_functions.``A001836`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Numbers \(n\) such that \(\phi(2n-1) < \phi(2n)\), where \(\phi\) is Euler’s totient function.

  Euler’s totient function is also known as euler_phi, euler_phi is a standard Sage function.

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A001836; a
      Numbers n such that phi(2n-1) < phi(2n), where phi is Euler's totient function A000010.
      sage: a.offset
      1
      sage: a(1)
      53
      sage: a(8)
      683
      sage: a(300)
      17798
      sage: a.list(12)
      [53, 83, 158, 263, 293, 368, 578, 683, 743, 788, 878, 893]
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer

  Compare: Searching Sloane’s online database... Numbers n such that phi(2n-1) phi(2n), where phi is Euler’s totient function A000010. [53, 83, 158, 263, 293, 368, 578, 683, 743, 788, 878, 893]

  AUTHORS:

  - Jaap Spies (2007-01-17)
  `list`(n)

    EXAMPLES:

        sage: sloane.A001836.list(9)
        [53, 83, 158, 263, 293, 368, 578, 683, 743]

class `sage.combinat.sloane_functions.``A001906`

  Bases: `sage.combinat.sloane_functions.RecurrenceSequence2`

  \(F(2n) =\) bisection of Fibonacci sequence: \(a(n)=3a(n-1)-a(n-2)\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A001906; a
      F(2n) = bisection of Fibonacci sequence: a(n)=3a(n-1)-a(n-2).
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(8)
      987
      sage: a(22)
      701408733
      sage: a.list(12)
      [0, 1, 3, 8, 21, 55, 144, 377, 987, 2584, 6765, 17711]

  AUTHORS:

  - Jaap Spies (2007-01-19)

class `sage.combinat.sloane_functions.``A001909`

  Bases: `sage.combinat.sloane_functions.ExtremesOfPermanentsSequence`

  \(a(n) = n*a(n-1) + (n-4)*a(n-2)\), with \(a(2) = 0\), \(a(3) = 1\).

  With offset 1, permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=4\) and \(n\) zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

  Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

  INPUT:

  - `n` - positive integer = 2

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A001909;a
      a(n) = n*a(n-1) + (n-4)*a(n-2), a(2) = 0, a(3) = 1.
      sage: a(1)
      Traceback (most recent call last):
      ...
      ValueError: input n (=1) must be an integer >= 2
      sage: a.offset
      2
      sage: a(2)
      0
      sage: a(8)
      8544
      sage: a(22)
      470033715095287415734
      sage: a.list(9)
      [0, 1, 4, 21, 134, 1001, 8544, 81901, 870274]

  AUTHORS:

  - Jaap Spies (2007-01-13)

class `sage.combinat.sloane_functions.``A001910`

  Bases: `sage.combinat.sloane_functions.ExtremesOfPermanentsSequence`

  \(a(n) = n*a(n-1) + (n-5)*a(n-2)\), with \(a(3) = 0\), \(a(4) = 1\).

  With offset 1, permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=5\) and \(n\) zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

  Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

  INPUT:

  - `n` - positive integer = 3

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A001910;a
      a(n) = n*a(n-1) + (n-5)*a(n-2), a(3) = 0, a(4) = 1.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be an integer >= 3
      sage: a(3)
      0
      sage: a.offset
      3
      sage: a(8)
      1909
      sage: a(22)
      98125321641110663023
      sage: a.list(9)
      [0, 1, 5, 31, 227, 1909, 18089, 190435, 2203319]

  AUTHORS:

  - Jaap Spies (2007-01-13)

class `sage.combinat.sloane_functions.``A001969`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Evil numbers: even number of 1’s in binary expansion.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A001969;a
      Evil numbers: even number of 1's in binary expansion.
      sage: a(0)
      0
      sage: a(1)
      3
      sage: a(2)
      5
      sage: a(12)
      24
      sage: a.list(13)
      [0, 3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 23, 24]

  AUTHORS:

  - Jaap Spies (2007-02-02)

class `sage.combinat.sloane_functions.``A002110`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Primorial numbers (first definition): product of first \(n\) primes. Sometimes written \(p\#\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A002110;a
      Primorial numbers (first definition): product of first n primes. Sometimes written p#.
      sage: a(0)
      1
      sage: a(2)
      6
      sage: a(8)
      9699690
      sage: a(17)
      1922760350154212639070
      sage: a.list(9)
      [1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690]

  AUTHORS:

  - Jaap Spies (2007-01-25)

class `sage.combinat.sloane_functions.``A002113`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Palindromes in base 10.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A002113;a
      Palindromes in base 10.
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(2)
      2
      sage: a(12)
      33
      sage: a.list(13)
      [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33]

  AUTHORS:

  - Jaap Spies (2007-02-02)
  `list`(n)

    EXAMPLES:

        sage: sloane.A002113.list(15)
        [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55]

class `sage.combinat.sloane_functions.``A002275`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Repunits: \(\frac {(10^n - 1)}{9}\). Often denoted by \(R_n\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A002275;a
      Repunits: (10^n - 1)/9. Often denoted by R_n.
      sage: a(0)
      0
      sage: a(2)
      11
      sage: a(8)
      11111111
      sage: a(20)
      11111111111111111111
      sage: a.list(9)
      [0, 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111]

  AUTHORS:

  - Jaap Spies (2007-01-25)

class `sage.combinat.sloane_functions.``A002378`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Oblong (or pronic, or heteromecic) numbers: \(n(n+1)\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A002378;a
      Oblong (or pronic, or heteromecic) numbers: n(n+1).
      sage: a(-1)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be an integer >= 0
      sage: a(0)
      0
      sage: a(1)
      2
      sage: a(11)
      132
      sage: a.list(12)
      [0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class `sage.combinat.sloane_functions.``A002620`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, \(\lfloor n^2/4 \rfloor\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A002620;a
      Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, floor(n^2/4).
      sage: a(0)
      0
      sage: a(1)
      0
      sage: a(2)
      1
      sage: a(10)
      25
      sage: a.list(12)
      [0, 0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class `sage.combinat.sloane_functions.``A002808`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  The composite numbers: numbers \(n\) of the form \(xy\) for \(x > 1\) and \(y > 1\).

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A002808;a
      The composite numbers: numbers n of the form x*y for x > 1 and y > 1.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(2)
      6
      sage: a(11)
      20
      sage: a.list(12)
      [4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21]

  AUTHORS:

  - Jaap Spies (2007-01-26)
  `list`(n)

    EXAMPLES:

        sage: sloane.A002808.list(10)
        [4, 6, 8, 9, 10, 12, 14, 15, 16, 18]

class `sage.combinat.sloane_functions.``A003418`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Least common multiple (or lcm) of \(\{1, 2, \cdots, n\}\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A003418;a
      Least common multiple (or lcm) of {1, 2, ..., n}.
      sage: a(0)
      1
      sage: a(1)
      1
      sage: a(13)
      360360
      sage: a.list(14)
      [1, 1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, 27720, 27720, 360360]
      sage: a(20.0)
      Traceback (most recent call last):
      ...
      TypeError: input must be an int, long, or Integer

  AUTHOR:

  - Jaap Spies (2007-01-31)

class `sage.combinat.sloane_functions.``A004086`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Read n backwards (referred to as \(R(n)\) in many sequences).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A004086;a
      Read n backwards (referred to as R(n) in many sequences).
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(2)
      2
      sage: a(3333)
      3333
      sage: a(12345)
      54321
      sage: a.list(13)
      [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 21]

  AUTHORS:

  - Jaap Spies (2007-02-02)

class `sage.combinat.sloane_functions.``A004526`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  The nonnegative integers repeated

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A004526;a
      The nonnegative integers repeated.
      sage: a(0)
      0
      sage: a(1)
      0
      sage: a(2)
      1
      sage: a(10)
      5
      sage: a.list(12)
      [0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class `sage.combinat.sloane_functions.``A005100`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Deficient numbers: \(\sigma(n) < 2n\).

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A005100;a
      Deficient numbers: sigma(n) < 2n
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      1
      sage: a(2)
      2
      sage: a(12)
      14
      sage: a.list(12)
      [1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14]

  AUTHORS:

  - Jaap Spies (2007-01-26)
  `list`(n)

    EXAMPLES:

        sage: sloane.A005100.list(10)
        [1, 2, 3, 4, 5, 7, 8, 9, 10, 11]

class `sage.combinat.sloane_functions.``A005101`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Abundant numbers (sum of divisors of \(n\) exceeds \(2n\)).

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A005101;a
      Abundant numbers (sum of divisors of n exceeds 2n).
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      12
      sage: a(2)
      18
      sage: a(12)
      60
      sage: a.list(12)
      [12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60]

  AUTHORS:

  - Jaap Spies (2007-01-26)
  `list`(n)

    EXAMPLES:

        sage: sloane.A005101.list(10)
        [12, 18, 20, 24, 30, 36, 40, 42, 48, 54]

class `sage.combinat.sloane_functions.``A005117`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Square-free numbers

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A005117;a
      Square-free numbers.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(2)
      2
      sage: a(12)
      17
      sage: a.list(12)
      [1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17]

  AUTHORS:

  - Jaap Spies (2007-01-25)
  `list`(n)

    EXAMPLES:

        sage: sloane.A005117.list(10)
        [1, 2, 3, 5, 6, 7, 10, 11, 13, 14]

class `sage.combinat.sloane_functions.``A005408`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  The odd numbers a(n) = 2n + 1.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A005408;a
      The odd numbers a(n) = 2n + 1.
      sage: a(-1)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be an integer >= 0
      sage: a(0)
      1
      sage: a(4)
      9
      sage: a(11)
      23
      sage: a.list(12)
      [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23]

  AUTHORS:

  - Jaap Spies (2007-01-26)

class `sage.combinat.sloane_functions.``A005843`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  The even numbers: \(a(n) = 2n\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A005843;a
      The even numbers: a(n) = 2n.
      sage: a(0.0)
      Traceback (most recent call last):
      ...
      TypeError: input must be an int, long, or Integer
      sage: a(1)
      2
      sage: a(2)
      4
      sage: a(9)
      18
      sage: a.list(10)
      [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]

  AUTHORS:

  - Jaap Spies (2007-02-03)

class `sage.combinat.sloane_functions.``A006318`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Large Schroeder numbers.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A006318;a
      Large Schroeder numbers.
      sage: a(0)
      1
      sage: a(1)
      2
      sage: a(2)
      6
      sage: a(9)
      206098
      sage: a.list(10)
      [1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098]

  AUTHORS:

  - Jaap Spies (2007-02-03)

class `sage.combinat.sloane_functions.``A006530`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Largest prime dividing \(n\) (with \(a(1)=1\)).

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A006530;a
      Largest prime dividing n (with a(1)=1).
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      1
      sage: a(2)
      2
      sage: a(8)
      2
      sage: a(11)
      11
      sage: a.list(15)
      [1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5]

  AUTHORS:

  - Jaap Spies (2007-01-25)

class `sage.combinat.sloane_functions.``A006882`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Double factorials \(n!!\): \(a(n)=n \cdot a(n-2)\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A006882;a
      Double factorials n!!: a(n)=n*a(n-2).
      sage: a(0)
      1
      sage: a(2)
      2
      sage: a(8)
      384
      sage: a(20)
      3715891200
      sage: a.list(9)
      [1, 1, 2, 3, 8, 15, 48, 105, 384]

  AUTHORS:

  - Jaap Spies (2007-01-24)
  `df`()

    Double factorials n!!: a(n)=n*a(n-2).

    EXAMPLES:

        sage: it = sloane.A006882.df()
        sage: [it.next() for i in range(10)]
        [1, 1, 2, 3, 8, 15, 48, 105, 384, 945]

  `list`(n)

    EXAMPLES:

        sage: sloane.A006882.list(10)
        [1, 1, 2, 3, 8, 15, 48, 105, 384, 945]

class `sage.combinat.sloane_functions.``A007318`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Pascal’s triangle read by rows: \(C(n,k) = {n \choose k} = \frac {n!} {(k!(n-k)!)}\), \(0 \le k \le n\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A007318
      sage: a(0)
      1
      sage: a(1)
      1
      sage: a(13)
      4
      sage: a.list(15)
      [1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1]
      sage: a(100)
      715

  AUTHORS:

  - Jaap Spies (2007-01-31)

class `sage.combinat.sloane_functions.``A008275`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Triangle of Stirling numbers of first kind, \(s(n,k)\), \(n \ge 1\), \(1 \le k \le n\).

  The unsigned numbers are also called Stirling cycle numbers:

  \(|s(n,k)|\) = number of permutations of \(n\) objects with exactly \(k\) cycles.

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A008275;a
      Triangle of Stirling numbers of first kind, s(n,k), n >= 1, 1<=k<=n.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      1
      sage: a(2)
      -1
      sage: a(3)
      1
      sage: a(11)
      24
      sage: a.list(12)
      [1, -1, 1, 2, -3, 1, -6, 11, -6, 1, 24, -50]

  AUTHORS:

  - Jaap Spies (2007-02-02)
  `s`(n, k)

    EXAMPLES:

        sage: sloane.A008275.s(4,2)
        11
        sage: sloane.A008275.s(5,2)
        -50
        sage: sloane.A008275.s(5,3)
        35

class `sage.combinat.sloane_functions.``A008277`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Triangle of Stirling numbers of 2nd kind, \(S2(n,k)\), \(n \ge 1\), \(1 \le k \le n\).

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A008277;a
      Triangle of Stirling numbers of 2nd kind, S2(n,k), n >= 1, 1<=k<=n.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      1
      sage: a(2)
      1
      sage: a(3)
      1
      sage: a(4.0)
      Traceback (most recent call last):
      ...
      TypeError: input must be an int, long, or Integer
      sage: a.list(15)
      [1, 1, 1, 1, 3, 1, 1, 7, 6, 1, 1, 15, 25, 10, 1]

  AUTHORS:

  - Jaap Spies (2007-01-31)
  `s2`(n, k)

    Returns the Stirling number S2(n,k) of the 2nd kind.

    EXAMPLES:

        sage: sloane.A008277.s2(4,2)
        7

class `sage.combinat.sloane_functions.``A008683`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Moebius function \(\mu(n)\).

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A008683;a
      Moebius function mu(n).
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(2)
      -1
      sage: a(12)
      0
      sage: a.list(12)
      [1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0]

  AUTHORS:

  - Jaap Spies (2007-01-13)

class `sage.combinat.sloane_functions.``A010060`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Thue-Morse sequence.

  Let \(A_k\) denote the first \(2^k\) terms; then \(A_0 = 0\), and for \(k \ge 0\), \(A_{k+1} = A_k B_k\), where \(B_k\) is obtained from \(A_k\) by interchanging 0’s and 1’s.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A010060;a
      Thue-Morse sequence.
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(2)
      1
      sage: a(12)
      0
      sage: a.list(13)
      [0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0]

  AUTHORS:

  - Jaap Spies (2007-02-02)

class `sage.combinat.sloane_functions.``A015521`

  Bases: `sage.combinat.sloane_functions.RecurrenceSequence2`

  Linear 2nd order recurrence, \(a(0)=0\), \(a(1)=1\) and \(a(n) = 3 a(n-1) + 4 a(n-2)\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A015521; a
      Linear 2nd order recurrence, a(n) = 3 a(n-1) + 4 a(n-2).
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(8)
      13107
      sage: a(41)
      967140655691703339764941
      sage: a.list(12)
      [0, 1, 3, 13, 51, 205, 819, 3277, 13107, 52429, 209715, 838861]

  AUTHORS:

  - Jaap Spies (2007-01-19)

class `sage.combinat.sloane_functions.``A015523`

  Bases: `sage.combinat.sloane_functions.RecurrenceSequence2`

  Linear 2nd order recurrence, \(a(0)=0\), \(a(1)=1\) and \(a(n) = 3 a(n-1) + 5 a(n-2)\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A015523; a
      Linear 2nd order recurrence, a(n) = 3 a(n-1) + 5 a(n-2).
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(8)
      17727
      sage: a(41)
      6173719566474529739091481
      sage: a.list(12)
      [0, 1, 3, 14, 57, 241, 1008, 4229, 17727, 74326, 311613, 1306469]

  AUTHORS:

  - Jaap Spies (2007-01-19)

class `sage.combinat.sloane_functions.``A015530`

  Bases: `sage.combinat.sloane_functions.RecurrenceSequence2`

  Linear 2nd order recurrence, \(a(0)=0\), \(a(1)=1\) and \(a(n) = 4 a(n-1) + 3 a(n-2)\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A015530;a
      Linear 2nd order recurrence, a(n) = 4 a(n-1) + 3 a(n-2).
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(2)
      4
      sage: a.offset
      0
      sage: a(8)
      41008
      sage: a.list(9)
      [0, 1, 4, 19, 88, 409, 1900, 8827, 41008]

  AUTHORS:

  - Jaap Spies (2007-01-19)

class `sage.combinat.sloane_functions.``A015531`

  Bases: `sage.combinat.sloane_functions.RecurrenceSequence2`

  Linear 2nd order recurrence, \(a(0)=0\), \(a(1)=1\) and \(a(n) = 4 a(n-1) + 5 a(n-2)\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A015531;a
      Linear 2nd order recurrence, a(n) = 4 a(n-1) + 5 a(n-2).
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(2)
      4
      sage: a.offset
      0
      sage: a(8)
      65104
      sage: a(60)
      144560289664733924534327040115992228190104
      sage: a.list(9)
      [0, 1, 4, 21, 104, 521, 2604, 13021, 65104]

  AUTHORS:

  - Jaap Spies (2007-01-19)

class `sage.combinat.sloane_functions.``A015551`

  Bases: `sage.combinat.sloane_functions.RecurrenceSequence2`

  Linear 2nd order recurrence, \(a(0)=0\), \(a(1)=1\) and \(a(n) = 6 a(n-1) + 5 a(n-2)\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A015551;a
      Linear 2nd order recurrence, a(n) = 6 a(n-1) + 5 a(n-2).
      sage: a(0)
      0
      sage: a(1)
      1
      sage: a(2)
      6
      sage: a.offset
      0
      sage: a(8)
      570216
      sage: a(60)
      7110606606530059736761484557155863822531970573036
      sage: a.list(9)
      [0, 1, 6, 41, 276, 1861, 12546, 84581, 570216]

  AUTHORS:

  - Jaap Spies (2007-01-19)

class `sage.combinat.sloane_functions.``A018252`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  The nonprime numbers, starting with 1.

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A018252;a
      The nonprime numbers.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      1
      sage: a(2)
      4
      sage: a(9)
      15
      sage: a.list(10)
      [1, 4, 6, 8, 9, 10, 12, 14, 15, 16]

  AUTHORS:

  - Jaap Spies (2007-02-04)

class `sage.combinat.sloane_functions.``A020639`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Least prime dividing \(n\) with \(a(1)=1\).

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A020639;a
      Least prime dividing n (a(1)=1).
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      1
      sage: a(13)
      13
      sage: a.list(14)
      [1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2]

  AUTHORS:

  - Jaap Spies (2007-01-25)
  `list`(n)

    EXAMPLES:

        sage: sloane.A020639.list(10)
        [1, 2, 3, 2, 5, 2, 7, 2, 3, 2]

class `sage.combinat.sloane_functions.``A046660`(offset=1)

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Excess of \(n\) = number of prime divisors (with multiplicity) - number of prime divisors (without multiplicity).

  \(\Omega(n) - \omega(n)\).

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A046660; a
      Excess of n = Bigomega (with multiplicity) - omega (without multiplicity).
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      0
      sage: a(8)
      2
      sage: a(41)
      0
      sage: a(84792)
      2
      sage: a.list(12)
      [0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1]

  AUTHORS:

  > - Jaap Spies (2007-01-19)

class `sage.combinat.sloane_functions.``A049310`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Triangle of coefficients of Chebyshev’s \(S(n,x)\): \(U(n, \frac x 2)\) polynomials (exponents in increasing order).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A049310;a
      Triangle of coefficients of Chebyshev's S(n,x) := U(n,x/2) polynomials (exponents in increasing order).
      sage: a(0)
      1
      sage: a(1)
      0
      sage: a(13)
      0
      sage: a.list(15)
      [1, 0, 1, -1, 0, 1, 0, -2, 0, 1, 1, 0, -3, 0, 1]
      sage: a(200)
      0
      sage: a.keyword
      ['sign', 'tabl', 'nice', 'easy', 'core', 'triangle']

  AUTHORS:

  - Jaap Spies (2007-01-31)

class `sage.combinat.sloane_functions.``A051959`

  Bases: `sage.combinat.sloane_functions.RecurrenceSequence`

  Linear second order recurrence. A051959.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A051959; a
      Linear second order recurrence. A051959.
      sage: a(0)
      1
      sage: a(1)
      10
      sage: a(8)
      9969
      sage: a(41)
      42834431872413650
      sage: a.list(12)
      [1, 10, 36, 104, 273, 686, 1688, 4112, 9969, 24114, 58268, 140728]

  AUTHORS:

  - Jaap Spies (2007-01-19)
  `g`(k)

    EXAMPLES:

        sage: sloane.A051959.g(2)
        15
        sage: sloane.A051959.g(1)
        0

class `sage.combinat.sloane_functions.``A055790`

  Bases: `sage.combinat.sloane_functions.ExtremesOfPermanentsSequence2`

  \(a(n) = n*a(n-1) + (n-2)*a(n-2) [a(0) = 0, a(1) = 2]\).

  With offset 1, permanent of (0,1)-matrix of size n X (n+d) with d=1 and n-1 zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

  REFERENCES:

  - Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A055790;a
      a(n) = n*a(n-1) + (n-2)*a(n-2) [a(0) = 0, a(1) = 2].
      sage: a(0)
      0
      sage: a(1)
      2
      sage: a(2)
      4
      sage: a.offset
      0
      sage: a(8)
      165016
      sage: a(22)
      10356214297533070441564
      sage: a.list(9)
      [0, 2, 4, 14, 64, 362, 2428, 18806, 165016]

  AUTHORS:

  - Jaap Spies (2007-01-23)

class `sage.combinat.sloane_functions.``A061084`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Fibonacci-type sequence based on subtraction: \(a(0) = 1\), \(a(1) = 2\) and \(a(n) = a(n-2)-a(n-1)\).

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A061084; a
      Fibonacci-type sequence based on subtraction: a(0) = 1, a(1) = 2 and a(n) = a(n-2)-a(n-1).
      sage: a(0)
      1
      sage: a(1)
      2
      sage: a(8)
      -29
      sage: a(22)
      -24476
      sage: a.list(12)
      [1, 2, -1, 3, -4, 7, -11, 18, -29, 47, -76, 123]
      sage: a.keyword
      ['sign', 'easy', 'nice']

  AUTHORS:

  - Jaap Spies (2007-01-18)

class `sage.combinat.sloane_functions.``A064553`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  \(a(1) = 1\), \(a(prime(i)) = i + 1\) for \(i > 0\) and \(a(u \cdot v) = a(u) \cdot a(v)\) for \(u, v > 0\).

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A064553;a
      a(1) = 1, a(prime(i)) = i+1 for i > 0 and a(u*v) = a(u)*a(v) for u,v > 0
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      1
      sage: a(2)
      2
      sage: a(9)
      9
      sage: a.list(16)
      [1, 2, 3, 4, 4, 6, 5, 8, 9, 8, 6, 12, 7, 10, 12, 16]

  AUTHORS:

  - Jaap Spies (2007-02-04)

class `sage.combinat.sloane_functions.``A079922`(offset=1)

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  function returns solutions to the Dancing School problem with \(n\) girls and \(n+3\) boys.

  The value is \(per(B)\), the permanent of the (0,1)-matrix \(B\) of size \(n \times n+3\) with \(b(i,j)=1\) if and only if \(i \le j \le i+n\).

  REFERENCES:

  - Jaap Spies, Nieuw Archief voor Wiskunde, 5/7 nr 4, December 2006

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A079922; a
      Solutions to the Dancing School problem with n girls and n+3 boys
      sage: a.offset
      1
      sage: a(1)
      4
      sage: a(8)
      2227
      sage: a.list(8)
      [4, 13, 36, 90, 212, 478, 1044, 2227]

  Compare: Searching Sloane’s online database... Solution to the Dancing School Problem with n girls and n+3 boys: f(n,3). [4, 13, 36, 90, 212, 478, 1044, 2227]

      sage: a(-1)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be a positive integer

  AUTHORS:

  > - Jaap Spies (2007-01-14)

class `sage.combinat.sloane_functions.``A079923`(offset=1)

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  function returns solutions to the Dancing School problem with \(n\) girls and \(n+4\) boys.

  The value is \(per(B)\), the permanent of the (0,1)-matrix \(B\) of size \(n \times n+3\) with \(b(i,j)=1\) if and only if \(i \le j \le i+n\).

  REFERENCES:

  - Jaap Spies, Nieuw Archief voor Wiskunde, 5/7 nr 4, December 2006

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A079923; a
      Solutions to the Dancing School problem with n girls and n+4 boys
      sage: a.offset
      1
      sage: a(1)
      5
      sage: a(8)
      15458
      sage: a.list(8)
      [5, 21, 76, 246, 738, 2108, 5794, 15458]

  Compare: Searching Sloane’s online database... Solution to the Dancing School Problem with n girls and n+4 boys: f(n,4). [5, 21, 76, 246, 738, 2108, 5794, 15458]

      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer

  AUTHORS:

  > - Jaap Spies (2007-01-17)

class `sage.combinat.sloane_functions.``A082411`

  Bases: `sage.combinat.sloane_functions.RecurrenceSequence2`

  Second-order linear recurrence sequence with \(a(n) = a(n-1) + a(n-2)\).

  \(a(0) = 407389224418\), \(a(1) = 76343678551\). This is the second-order linear recurrence sequence with \(a(0)\) and \(a(1)\) co-prime, that R. L. Graham in 1964 stated did not contain any primes.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A082411;a
      Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
      sage: a(1)
      76343678551
      sage: a(2)
      483732902969
      sage: a(3)
      560076581520
      sage: a(20)
      2219759332689173
      sage: a.list(4)
      [407389224418, 76343678551, 483732902969, 560076581520]

  AUTHORS:

  - Jaap Spies (2007-01-23)

class `sage.combinat.sloane_functions.``A083103`

  Bases: `sage.combinat.sloane_functions.RecurrenceSequence2`

  Second-order linear recurrence sequence with \(a(n) = a(n-1) + a(n-2)\).

  \(a(0) = 1786772701928802632268715130455793\), \(a(1) = 1059683225053915111058165141686995\). This is the second-order linear recurrence sequence with \(a(0)\) and \(a(1)\) co- prime, that R. L. Graham in 1964 stated did not contain any primes. It has not been verified. Graham made a mistake in the calculation that was corrected by D. E. Knuth in 1990.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A083103;a
      Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
      sage: a(1)
      1059683225053915111058165141686995
      sage: a(2)
      2846455926982717743326880272142788
      sage: a(3)
      3906139152036632854385045413829783
      sage: a.offset
      0
      sage: a(8)
      45481392851206651551714764671352204
      sage: a(20)
      14639253684254059531823985143948191708
      sage: a.list(4)
      [1786772701928802632268715130455793, 1059683225053915111058165141686995, 2846455926982717743326880272142788, 3906139152036632854385045413829783]

  AUTHORS:

  - Jaap Spies (2007-01-23)

class `sage.combinat.sloane_functions.``A083104`

  Bases: `sage.combinat.sloane_functions.RecurrenceSequence2`

  Second-order linear recurrence sequence with \(a(n) = a(n-1) + a(n-2)\).

  \(a(0) = 331635635998274737472200656430763\), \(a(1) = 1510028911088401971189590305498785\). This is the second-order linear recurrence sequence with \(a(0)\) and \(a(1)\) co-prime. It was found by Ronald Graham in 1990.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A083104;a
      Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
      sage: a(3)
      3351693458175078679851381267428333
      sage: a.offset
      0
      sage: a(8)
      36021870400834012982120004949074404
      sage: a(20)
      11601914177621826012468849361236300628

  AUTHORS:

  - Jaap Spies (2007-01-23)

class `sage.combinat.sloane_functions.``A083105`

  Bases: `sage.combinat.sloane_functions.RecurrenceSequence2`

  Second-order linear recurrence sequence with \(a(n) = a(n-1) + a(n-2)\).

  \(a(0) = 62638280004239857\), \(a(1) = 49463435743205655\). This is the second-order linear recurrence sequence with \(a(0)\) and \(a(1)\) co-prime. It was found by Donald Knuth in 1990.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A083105;a
      Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
      sage: a(1)
      49463435743205655
      sage: a(2)
      112101715747445512
      sage: a(3)
      161565151490651167
      sage: a.offset
      0
      sage: a(8)
      1853029790662436896
      sage: a(20)
      596510791500513098192
      sage: a.list(4)
      [62638280004239857, 49463435743205655, 112101715747445512, 161565151490651167]

  AUTHORS:

  - Jaap Spies (2007-01-23)

class `sage.combinat.sloane_functions.``A083216`

  Bases: `sage.combinat.sloane_functions.RecurrenceSequence2`

  Second-order linear recurrence sequence with \(a(n) = a(n-1) + a(n-2)\).

  \(a(0) = 20615674205555510\), \(a(1) = 3794765361567513\). This is a second-order linear recurrence sequence with \(a(0)\) and \(a(1)\) co-prime that does not contain any primes. It was found by Herbert Wilf in 1990.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A083216; a
      Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
      sage: a(0)
      20615674205555510
      sage: a(1)
      3794765361567513
      sage: a(8)
      347693837265139403
      sage: a(41)
      2738025383211084205003383
      sage: a.list(4)
      [20615674205555510, 3794765361567513, 24410439567123023, 28205204928690536]

  AUTHORS:

  - Jaap Spies (2007-01-19)

class `sage.combinat.sloane_functions.``A090010`

  Bases: `sage.combinat.sloane_functions.ExtremesOfPermanentsSequence2`

  Permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=6\) and \(n\) zeros not on a line.

  ` a(n) = (n+5)*a(n-1) + (n-1)*a(n-2), a(1)=6, a(2)=43`.

  This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

  REFERENCES:

  - Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A090010;a
      Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n zeros not on a line.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      6
      sage: a(2)
      43
      sage: a.offset
      1
      sage: a(8)
      67741129
      sage: a(22)
      192416593029158989003270143
      sage: a.list(9)
      [6, 43, 356, 3333, 34754, 398959, 4996032, 67741129, 988344062]

  AUTHORS:

  - Jaap Spies (2007-01-23)

class `sage.combinat.sloane_functions.``A090012`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=2\) and \(n-1\) zeros not on a line.

  \(a(n) = (n+1)*a(n-1) + (n-2)*a(n-2)\), \(a(1)=3\) and \(a(2)=9\)

  This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

  REFERENCES:

  - Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A090012;a
      Permanent of (0,1)-matrix of size n X (n+d) with d=2 and n-1 zeros not on a line.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      3
      sage: a(2)
      9
      sage: a.offset
      1
      sage: a(8)
      890901
      sage: a(22)
      129020386652297208795129
      sage: a.list(9)
      [3, 9, 39, 213, 1395, 10617, 91911, 890901, 9552387]

  AUTHORS:

  - Jaap Spies (2007-01-23)

class `sage.combinat.sloane_functions.``A090013`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=3\) and \(n-1\) zeros not on a line.

  \(a(n) = (n+1)*a(n-1) + (n-2)*a(n-2) [a(1)=4, a(2)=16]\)

  This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

  REFERENCES:

  - Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A090013;a
      Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      4
      sage: a(2)
      16
      sage: a.offset
      1
      sage: a(8)
      3481096
      sage: a(22)
      1112998577171142607670336
      sage: a.list(9)
      [4, 16, 84, 536, 4004, 34176, 327604, 3481096, 40585284]

  AUTHORS:

  - Jaap Spies (2007-01-23)

class `sage.combinat.sloane_functions.``A090014`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=4\) and \(n-1\) zeros not on a line.

  \(a(n) = (n+1)*a(n-1) + (n-2)*a(n-2) [a(1)=5, a(2)=25]\)

  This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

  REFERENCES:

  - Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A090014;a
      Permanent of (0,1)-matrix of size n X (n+d) with d=4 and n-1 zeros not on a line.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      5
      sage: a(2)
      25
      sage: a.offset
      1
      sage: a(8)
      11016595
      sage: a(22)
      7469733600354446865509725
      sage: a.list(9)
      [5, 25, 155, 1135, 9545, 90445, 952175, 11016595, 138864365]

  AUTHORS:

  - Jaap Spies (2007-01-23)

class `sage.combinat.sloane_functions.``A090015`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=5\) and \(n-1\) zeros not on a line.

  \(a(n) = (n+1)*a(n-1) + (n-2)*a(n-2) [a(1)=6, a(2)=36]\)

  This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

  REFERENCES:

  - Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A090015;a
      Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a line.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      6
      sage: a(2)
      36
      sage: a.offset
      1
      sage: a(8)
      29976192
      sage: a(22)
      41552258517692116794936876
      sage: a.list(9)
      [6, 36, 258, 2136, 19998, 208524, 2393754, 29976192, 406446774]

  AUTHORS:

  - Jaap Spies (2007-01-23)

class `sage.combinat.sloane_functions.``A090016`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Permanent of (0,1)-matrix of size \(n \times (n+d)\) with \(d=6\) and \(n-1\) zeros not on a line.

  \(a(n) = (n+1)*a(n-1) + (n-2)*a(n-2) [a(1)=7, a(2)=49]\)

  \(A090016 a(n) = A090010(n-1) + A090010(n), a(1)=7\)

  This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

  REFERENCES:

  - Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A090016;a
      Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n-1 zeros not on a line.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(1)
      7
      sage: a(2)
      49
      sage: a.offset
      1
      sage: a(8)
      72737161
      sage: a(22)
      199341969448774341802426289
      sage: a.list(9)
      [7, 49, 399, 3689, 38087, 433713, 5394991, 72737161, 1056085191]

  AUTHORS:

  - Jaap Spies (2007-01-23)

class `sage.combinat.sloane_functions.``A109814`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  The \(n\) th term of the sequence \(a(n)\) is the largest \(k\) such that \(n\) can be written as sum of \(k\) consecutive integers.

  By definition, \(n\) is the sum of at most \(a(n)\) consecutive positive integers. Suppose \(n\) is to be written as sum of \(k\) consecutive integers starting with \(m\), then \(2n = k(2m + k - 1)\). Only one of the factors is odd. For each odd divisor \(d\) of \(n\) there is a unique corresponding \(k = min(d,2n/d)\). \(a(n)\) can be alternatively defined as the largest among those \(k\) .

  See also

  - Wikipedia article on polite numbers.
  - An exercise sheet (with answers) about sums of consecutive integers.

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A109814; a
      a(n) is the largest k such that n can be written as sum of k consecutive positive integers.
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(2)
      1
      sage: a.list(9)
      [1, 1, 2, 1, 2, 3, 2, 1, 3]

  AUTHORS:

  - Jaap Spies (2007-01-13)

class `sage.combinat.sloane_functions.``A111774`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Sequence of numbers of the third kind, i.e., numbers that can be written as a sum of at least three consecutive positive integers.

  Odd primes can only be written as a sum of two consecutive integers. Powers of 2 do not have a representation as a sum of \(k\) consecutive integers (other than the trivial \(n = n\) for \(k = 1\)).

  See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A111774; a
      Numbers that can be written as a sum of at least three consecutive positive integers.
      sage: a(1)
      6
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(100)
      141
      sage: a(156)
      209
      sage: a(302)
      386
      sage: a.list(12)
      [6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25]
      sage: a(1/3)
      Traceback (most recent call last):
      ...
      TypeError: input must be an int, long, or Integer

  AUTHORS:

  - Jaap Spies (2007-01-13)
  `is_number_of_the_third_kind`(n)

    This function returns True if and only if \(n\) is a number of the third kind.

    A number is of the third kind if it can be written as a sum of at least three consecutive positive integers. Odd primes can only be written as a sum of two consecutive integers. Powers of 2 do not have a representation as a sum of \(k\) consecutive integers (other than the trivial \(n = n\) for \(k = 1\)).

    See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf

    INPUT:

    - `n` - positive integer

    OUTPUT:

    - `True` - if n is not prime and not a power of 2 False -

    EXAMPLES:

        sage: a = sloane.A111774
        sage: a.is_number_of_the_third_kind(6)
        True
        sage: a.is_number_of_the_third_kind(100)
        True
        sage: a.is_number_of_the_third_kind(16)
        False
        sage: a.is_number_of_the_third_kind(97)
        False

    AUTHORS:

    - Jaap Spies (2006-12-09)
  `list`(n)

    EXAMPLES:

        sage: sloane.A111774.list(12)
        [6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25]

class `sage.combinat.sloane_functions.``A111775`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  Number of ways \(n\) can be written as a sum of at least three consecutive integers.

  Powers of 2 and (odd) primes can not be written as a sum of at least three consecutive integers. \(a(n)\) strongly depends on the number of odd divisors of \(n\) (A001227): Suppose \(n\) is to be written as sum of \(k\) consecutive integers starting with \(m\), then \(2n = k(2m + k - 1)\). Only one of the factors is odd. For each odd divisor of \(n\) there is a unique corresponding \(k\), \(k=1\) and \(k=2\) must be excluded.

  See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf

  INPUT:

  - `n` - non negative integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A111775; a
      Number of ways n can be written as a sum of at least three consecutive integers.

      sage: a(1)
      0
      sage: a(0)
      0

  We have a(15)=2 because 15 = 4+5+6 and 15 = 1+2+3+4+5. The number of odd divisors of 15 is 4.

      sage: a(15)
      2

      sage: a(100)
      2
      sage: a(256)
      0
      sage: a(29)
      0
      sage: a.list(20)
      [0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 2, 0, 0, 2, 0]
      sage: a(1/3)
      Traceback (most recent call last):
      ...
      TypeError: input must be an int, long, or Integer

  AUTHORS:

  - Jaap Spies (2006-12-09)

class `sage.combinat.sloane_functions.``A111787`

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  This function returns the \(n\)-th number of Sloane’s sequence A111787

  \(a(n)=0\) if \(n\) is an odd prime or a power of 2. For numbers of the third kind (see A111774) we proceed as follows: suppose \(n\) is to be written as sum of \(k\) consecutive integers starting with \(m\), then \(2n = k(2m + k - 1)\). Let \(p\) be the smallest odd prime divisor of \(n\) then \(a(n) = min(p,2n/p)\).

  See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf

  INPUT:

  - `n` - positive integer

  OUTPUT:

  - `integer` - function value

  EXAMPLES:

      sage: a = sloane.A111787; a
      a(n) is the least k >= 3 such that n can be written as sum of k consecutive integers. a(n)=0 if such a k does not exist.
      sage: a.offset
      1
      sage: a(1)
      0
      sage: a(0)
      Traceback (most recent call last):
      ...
      ValueError: input n (=0) must be a positive integer
      sage: a(100)
      5
      sage: a(256)
      0
      sage: a(29)
      0
      sage: a.list(20)
      [0, 0, 0, 0, 0, 3, 0, 0, 3, 4, 0, 3, 0, 4, 3, 0, 0, 3, 0, 5]
      sage: a(-1)
      Traceback (most recent call last):
      ...
      ValueError: input n (=-1) must be a positive integer

  AUTHORS:

  - Jaap Spies (2007-01-14)

class `sage.combinat.sloane_functions.``ExponentialNumbers`(a)

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  A sequence of Exponential numbers.

  EXAMPLES:

      sage: from sage.combinat.sloane_functions import ExponentialNumbers
      sage: ExponentialNumbers(0)
      Sequence of Exponential numbers around 0

class `sage.combinat.sloane_functions.``ExtremesOfPermanentsSequence`(offset=1)

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  A sequence starting at offset (=1 by default).

  EXAMPLES:

      sage: from sage.combinat.sloane_functions import SloaneSequence
      sage: SloaneSequence().offset
      1
      sage: SloaneSequence(4).offset
      4

  `gen`(a0, a1, d)

    EXAMPLES:

        sage: it = sloane.A000153.gen(0,1,2)
        sage: [it.next() for i in range(5)]
        [0, 1, 2, 7, 32]

  `list`(n)

    EXAMPLES:

        sage: sloane.A000153.list(8)
        [0, 1, 2, 7, 32, 181, 1214, 9403]

class `sage.combinat.sloane_functions.``ExtremesOfPermanentsSequence2`(offset=1)

  Bases: `sage.combinat.sloane_functions.ExtremesOfPermanentsSequence`

  A sequence starting at offset (=1 by default).

  EXAMPLES:

      sage: from sage.combinat.sloane_functions import SloaneSequence
      sage: SloaneSequence().offset
      1
      sage: SloaneSequence(4).offset
      4

  `gen`(a0, a1, d)

    EXAMPLES:

        sage: from sage.combinat.sloane_functions import ExtremesOfPermanentsSequence2
        sage: e = ExtremesOfPermanentsSequence2()
        sage: it = e.gen(6,43,6)
        sage: [it.next() for i in range(5)]
        [6, 43, 307, 2542, 23799]

class `sage.combinat.sloane_functions.``RecurrenceSequence`(offset=1)

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  A sequence starting at offset (=1 by default).

  EXAMPLES:

      sage: from sage.combinat.sloane_functions import SloaneSequence
      sage: SloaneSequence().offset
      1
      sage: SloaneSequence(4).offset
      4

  `list`(n)

    EXAMPLES:

        sage: sloane.A001110.list(8)
        [0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881]

class `sage.combinat.sloane_functions.``RecurrenceSequence2`(offset=1)

  Bases: `sage.combinat.sloane_functions.SloaneSequence`

  A sequence starting at offset (=1 by default).

  EXAMPLES:

      sage: from sage.combinat.sloane_functions import SloaneSequence
      sage: SloaneSequence().offset
      1
      sage: SloaneSequence(4).offset
      4

  `list`(n)

    EXAMPLES:

        sage: sloane.A001906.list(10)
        [0, 1, 3, 8, 21, 55, 144, 377, 987, 2584]

class `sage.combinat.sloane_functions.``Sloane`

  Bases: `sage.structure.sage_object.SageObject`

  A collection of Sloane generating functions.

  This class inspects sage.combinat.sloane_functions, accumulating all the SloaneSequence classes starting with ‘A’. These are listed for tab completion, but not instantiated until requested.

  EXAMPLES: Ensure we have lots of entries:

      sage: len(sloane.trait_names()) > 100
      True

  And ensure none are being incorrectly returned:

      sage: [ None for n in sloane.trait_names() if not n.startswith('A') ]
      []

  Ensure we can access dynamic constructions and cache correctly:

      sage: s = sloane.A000587
      sage: s is sloane.A000587
      True

  And that we can access other functions in parent classes:

      sage: sloane.__class__
      <class 'sage.combinat.sloane_functions.Sloane'>

  AUTHORS:

  - Nick Alexander
  `trait_names`()

    List Sloane generating functions for tab-completion. The member classes are inspected from module sage.combinat.sloane_functions.

    They must be sub classes of SloaneSequence and must start with ‘A’. These restrictions are only to prevent typos, incorrect inspecting, etc.

    EXAMPLES:

        sage: type(sloane.trait_names())
        <type 'list'>

class `sage.combinat.sloane_functions.``SloaneSequence`(offset=1)

  Bases: `sage.structure.sage_object.SageObject`

  Base class for a Sloane integer sequence.

  EXAMPLES:

  We create a dummy sequence:

  `list`(n)

    Return n terms of the sequence: sequence[offset], sequence[offset+1], ... , sequence[offset+n-1]. EXAMPLES:

        sage: sloane.A000012.list(4)
        [1, 1, 1, 1]

`sage.combinat.sloane_functions.``perm_mh`(m, h)

  This functions calculates \(f(g,h)\) from Sloane’s sequences A079908-A079928

  INPUT:

  - `m` - positive integer
  - `h` - non negative integer

  OUTPUT: permanent of the m x (m+h) matrix, etc.

  EXAMPLES:

      sage: from sage.combinat.sloane_functions import perm_mh
      sage: perm_mh(3,3)
      36
      sage: perm_mh(3,4)
      76

  AUTHORS:

  - Jaap Spies (2006)

`sage.combinat.sloane_functions.``recur_gen2`(a0, a1, a2, a3)

  homogeneous general second-order linear recurrence generator with fixed coefficients

  a(0) = a0, a(1) = a1, a(n) = a2*a(n-1) + a3*a(n-2)

  EXAMPLES:

      sage: from sage.combinat.sloane_functions import recur_gen2
      sage: it = recur_gen2(1,1,1,1)
      sage: [it.next() for i in range(10)]
      [1, 1, 2, 3, 5, 8, 13, 21, 34, 55]

`sage.combinat.sloane_functions.``recur_gen2b`(a0, a1, a2, a3, b)

  inhomogenous second-order linear recurrence generator with fixed coefficients and \(b = f(n)\)

  \(a(0) = a0\), \(a(1) = a1\), \(a(n) = a2*a(n-1) + a3*a(n-2) +f(n)\).

  EXAMPLES:

      sage: from sage.combinat.sloane_functions import recur_gen2b
      sage: it = recur_gen2b(1,1,1,1, lambda n: 0)
      sage: [it.next() for i in range(10)]
      [1, 1, 2, 3, 5, 8, 13, 21, 34, 55]

`sage.combinat.sloane_functions.``recur_gen3`(a0, a1, a2, a3, a4, a5)

  homogeneous general third-order linear recurrence generator with fixed coefficients

  a(0) = a0, a(1) = a1, a(2) = a2, a(n) = a3*a(n-1) + a4*a(n-2) + a5*a(n-3)

  EXAMPLES:

      sage: from sage.combinat.sloane_functions import recur_gen3
      sage: it = recur_gen3(1,1,1,1,1,1)
      sage: [it.next() for i in range(10)]
      [1, 1, 1, 3, 5, 9, 17, 31, 57, 105]

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