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# The On-Line Encyclopedia of Integer Sequences (OEIS)

You can query the OEIS (Online Database of Integer Sequences) through Sage in order to:

> - identify a sequence from its first terms.
> - obtain more terms, formulae, references, etc. for a given sequence.

AUTHORS:

> - Thierry Monteil (2012-02-10 – 2013-06-21): initial version.

EXAMPLES:

    sage: oeis
    The On-Line Encyclopedia of Integer Sequences (http://oeis.org/)

What about a sequence starting with \(3, 7, 15, 1\) ?

    sage: search = oeis([3, 7, 15, 1], max_results=4) ; search  # optional -- internet
    0: A001203: Continued fraction expansion of Pi.
    1: A165416: Irregular array read by rows: The n-th row contains those distinct positive integers that each, when written in binary, occurs as a substring in binary n.
    2: A193583: Number of fixed points under iteration of sum of squares of digits in base b.
    3: A082495: (2^n-1) mod n.

    sage: c = search[0] ; c                             # optional -- internet
    A001203: Continued fraction expansion of Pi.

    sage: c.first_terms(15)                             # optional -- internet
    (3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1)

    sage: c.examples()                                  # optional -- internet
    0: Pi = 3.1415926535897932384...
    1:    = 3 + 1/(7 + 1/(15 + 1/(1 + 1/(292 + ...))))
    2:    = [a_0; a_1, a_2, a_3, ...] = [3; 7, 15, 292, ...]

    sage: c.comments()                                  # optional -- internet
    0: The first 5,821,569,425 terms were computed by _Eric W. Weisstein_ on Sep 18 2011.
    1: The first 10,672,905,501 terms were computed by _Eric W. Weisstein_ on Jul 17 2013.
    2: The first 15,000,000,000 terms were computed by _Eric W. Weisstein_ on Jul 27 2013.

    sage: x = c.natural_object() ; x.parent()           # optional -- internet
    Field of all continued fractions

    sage: x.convergents()[:7]                           # optional -- internet
    [3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, 208341/66317]

    sage: RR(x.value())                                 # optional -- internet
    3.14159265358979
    sage: RR(x.value()) == RR(pi)                       # optional -- internet
    True

What about posets ? Are they hard to count ? To which other structures are they related ?

    sage: [Posets(i).cardinality() for i in range(10)]
    [1, 1, 2, 5, 16, 63, 318, 2045, 16999, 183231]
    sage: oeis(_)                                       # optional -- internet
    0: A000112: Number of partially ordered sets ("posets") with n unlabeled elements.
    sage: p = _[0]                                      # optional -- internet

    sage: 'hard' in p.keywords()                        # optional -- internet
    True
    sage: len(p.formulas())                             # optional -- internet
    0
    sage: len(p.first_terms())                          # optional -- internet
    17

    sage: p.cross_references(fetch=True)                # optional -- internet
    0: A000798: Number of different quasi-orders (or topologies, or transitive digraphs) with n labeled elements.
    1: A001035: Number of partially ordered sets ("posets") with n labeled elements (or labeled acyclic transitive digraphs).
    2: A001930: Number of topologies, or transitive digraphs with n unlabeled nodes.
    3: A006057: Number of labeled topologies with n points.
    4: A079263: Number of constrained mixed models with n factors.
    5: A079265: Number of antisymmetric transitive binary relations on n unlabeled points.

What does the Taylor expansion of the \(e^(e^x-1)`\) function have to do with primes ?

    sage: x = var('x') ; f(x) = e^(e^x - 1)
    sage: L = [a*factorial(b) for a,b in taylor(f(x), x, 0, 20).coeffs()] ; L
    [1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597,
    27644437, 190899322, 1382958545, 10480142147, 82864869804, 682076806159,
    5832742205057, 51724158235372]

    sage: oeis(L)                                       # optional -- internet
    0: A000110: Bell or exponential numbers: ways of placing n labeled balls into n indistinguishable boxes.

    sage: b = _[0]                                      # optional -- internet

    sage: b.formulas()[0]                               # optional -- internet
    'E.g.f.: exp( exp(x) - 1).'

    sage: b.comments()[89]                              # optional -- internet
    'Number n is prime if mod(a(n)-2,n) = 0. [From _Dmitry Kruchinin_, Feb 14 2012]'

    sage: [n for n in range(2, 20) if (b(n)-2) % n == 0]    # optional -- internet
    [2, 3, 5, 7, 11, 13, 17, 19]

See also

- If you plan to do a lot of automatic searches for subsequences, you should consider installing `SloaneEncyclopedia`, a local partial copy of the OEIS.
- Some infinite OEIS sequences are implemented in Sage, via the `sloane_functions` module.

Todo

- in case of flood, suggest the user to install the off-line database instead.
- interface with the off-line database (or reimplement it).

## Classes and methods

class `sage.databases.oeis.``FancyTuple`

  Bases: `tuple`

  This class inherits from `tuple`, it allows to nicely print tuples whose elements have a one line representation.

  EXAMPLES:

      sage: from sage.databases.oeis import FancyTuple
      sage: t = FancyTuple(['zero', 'one', 'two', 'three', 4]) ; t
      0: zero
      1: one
      2: two
      3: three
      4: 4

      sage: t[2]
      'two'

class `sage.databases.oeis.``OEIS`

  The On-Line Encyclopedia of Integer Sequences.

  `OEIS` is a class representing the On-Line Encyclopedia of Integer Sequences. You can query it using its methods, but `OEIS` can also be called directly with three arguments:

  - `query` - it can be:
    - a string representing an OEIS ID (e.g. ‘A000045’).
    - an integer representing an OEIS ID (e.g. 45).
    - a list representing a sequence of integers.
    - a string, representing a text search.
  - `max_results` - (integer, default: 30) the maximum number of results to return, they are sorted according to their relevance. In any cases, the OEIS website will never provide more than 100 results.
  - `first_result` - (integer, default: 0) allow to skip the `first_result` first results in the search, to go further. This is useful if you are looking for a sequence that may appear after the 100 first found sequences.

  OUTPUT:

  - if `query` is an integer or an OEIS ID (e.g. ‘A000045’), returns the associated OEIS sequence.
  - if `query` is a string, returns a tuple of OEIS sequences whose description corresponds to the query. Those sequences can be used without the need to fetch the database again.
  - if `query` is a list of integers, returns a tuple of OEIS sequences containing it as a subsequence. Those sequences can be used without the need to fetch the database again.

  EXAMPLES:

      sage: oeis
      The On-Line Encyclopedia of Integer Sequences (http://oeis.org/)

  A particular sequence can be called by its A-number or number:

      sage: oeis('A000040')                           # optional -- internet
      A000040: The prime numbers.

      sage: oeis(45)                                  # optional -- internet
      A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

  The database can be searched by subsequence:

      sage: search = oeis([1,2,3,5,8,13]) ; search    # optional -- internet
      0: A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.
      1: A027926: Triangular array T read by rows: T(n,0)=T(n,2n)=1 for n >= 0; ...
      2: A001129: Iccanobif numbers: reverse digits of two previous terms and add.

      sage: fibo = search[0]                         # optional -- internet

      sage: fibo.name()                               # optional -- internet
      'Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.'

      sage: fibo.first_terms()                        # optional -- internet
      (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987,
      1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393,
      196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887,
      9227465, 14930352, 24157817, 39088169)

      sage: fibo.cross_references()[0]                # optional -- internet
      'A039834'

      sage: fibo == oeis(45)                          # optional -- internet
      True

      sage: sfibo = oeis('A039834')                   # optional -- internet
      sage: sfibo.first_terms()                       # optional -- internet
      (1, 1, 0, 1, -1, 2, -3, 5, -8, 13, -21, 34, -55, 89, -144, 233,
      -377, 610, -987, 1597, -2584, 4181, -6765, 10946, -17711, 28657,
      -46368, 75025, -121393, 196418, -317811, 514229, -832040, 1346269,
      -2178309, 3524578, -5702887, 9227465, -14930352, 24157817)

      sage: sfibo.first_terms(absolute_value=True)[2:20] == fibo.first_terms()[:18]   # optional -- internet
      True

      sage: fibo.formulas()[3]                        # optional -- internet
      'F(n) = F(n-1) + F(n-2) = -(-1)^n F(-n).'

      sage: fibo.comments()[1]                        # optional -- internet
      "F(n+2) = number of binary sequences of length n that have no
      consecutive 0's."

      sage: fibo.links()[0]                           # optional -- internet
      'http://oeis.org/A000045/b000045.txt'

  The database can be searched by description:

      sage: oeis('prime gap factorization', max_results=4)                # optional -- internet
      0: A073491: Numbers having no prime gaps in their factorization.
      1: A073490: Number of prime gaps in factorization of n.
      2: A073492: Numbers having at least one prime gap in their factorization.
      3: A073493: Numbers having exactly one prime gap in their factorization.

  Warning

  The following will fetch the OEIS database twice (once for searching the database, and once again for creating the sequence `fibo`):

      sage: oeis([1,2,3,5,8,13])                  # optional -- internet
      0: A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.
      1: A027926: Triangular array T read by rows: T(n,0)=T(n,2n)=1 for n >= 0; ...
      2: A001129: Iccanobif numbers: reverse digits of two previous terms and add.

      sage: fibo = oeis('A000045')                # optional -- internet

  Do not do this, it is slow, it costs bandwidth and server resources ! Instead, do the following, to reuse the result of the search to create the sequence:

      sage: oeis([1,2,3,5,8,13])                  # optional -- internet
      0: A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.
      1: A027926: Triangular array T read by rows: T(n,0)=T(n,2n)=1 for n >= 0; ...
      2: A001129: Iccanobif numbers: reverse digits of two previous terms and add.

      sage: fibo = _[0]                           # optional -- internet

  `browse`()

    Open the OEIS web page in a browser.

    EXAMPLES:

        sage: oeis.browse()                         # optional -- webbrowser

  `find_by_description`(description, max_results=3, first_result=0)

    Search for OEIS sequences corresponding to the description.

    INPUT:

    - `description` - (string) the description the searched sequences.
    - `max_results` - (integer, default: 3) the maximum number of results we want. In any case, the on-line encyclopedia will not return more than 100 results.
    - `first_result` - (integer, default: 0) allow to skip the `first_result` first results in the search, to go further. This is useful if you are looking for a sequence that may appear after the 100 first found sequences.

    OUTPUT:

    - a tuple (with fancy formatting) of at most `max_results` OEIS sequences. Those sequences can be used without the need to fetch the database again.

    EXAMPLES:

        sage: oeis.find_by_description('prime gap factorization')       # optional -- internet
        0: A073491: Numbers having no prime gaps in their factorization.
        1: A073490: Number of prime gaps in factorization of n.
        2: A073492: Numbers having at least one prime gap in their factorization.

        sage: prime_gaps = _[1] ; prime_gaps        # optional -- internet
        A073490: Number of prime gaps in factorization of n.

        sage: oeis('beaver')                        # optional -- internet
        0: A028444: Busy Beaver sequence, or Rado's sigma function: ...
        1: A060843: Busy Beaver problem: a(n) = maximal number of steps ...
        2: A131956: Busy Beaver variation: maximum number of steps for ...

        sage: oeis('beaver', max_results=4, first_result=2)     # optional -- internet
        0: A131956: Busy Beaver variation: maximum number of steps for ...
        1: A141475: Number of Turing machines with n states following ...
        2: A131957: Busy Beaver sigma variation: maximum number of 1's ...
        3: A052200: Number of n-state, 2-symbol, d+ in {LEFT, RIGHT}, ...

  `find_by_id`(ident)

    INPUT:

    - `ident` - a string representing the A-number of the sequence or an integer representing its number.

    OUTPUT:

    - The OEIS sequence whose A-number or number corresponds to `ident`.

    EXAMPLES:

        sage: oeis.find_by_id('A000040')            # optional -- internet
        A000040: The prime numbers.

        sage: oeis.find_by_id(40)                   # optional -- internet
        A000040: The prime numbers.

  `find_by_subsequence`(subsequence, max_results=3, first_result=0)

    Search for OEIS sequences containing the given subsequence.

    INPUT:

    - `subsequence` - a list of integers.
    - `max_results` - (integer, default: 3), the maximum of results requested.
    - `first_result` - (integer, default: 0) allow to skip the `first_result` first results in the search, to go further. This is useful if you are looking for a sequence that may appear after the 100 first found sequences.

    OUTPUT:

    - a tuple (with fancy formatting) of at most `max_results` OEIS sequences. Those sequences can be used without the need to fetch the database again.

    EXAMPLES:

        sage: oeis.find_by_subsequence([2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377]) # optional -- internet
        0: A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.
        1: A177194: Fibonacci numbers whose decimal expression does not contain any digit 0.
        2: A020695: Pisot sequence E(2,3).

        sage: fibo = _[0] ; fibo                    # optional -- internet
        A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

class `sage.databases.oeis.``OEISSequence`(entry)

  Bases: `sage.structure.sage_object.SageObject`

  The class of OEIS sequences.

  This class implements OEIS sequences. Such sequences are produced from a string in the OEIS format. They are usually produced by calls to the On-Line Encyclopedia of Integer Sequences, represented by the class `OEIS`.

  Note

  Since some sequences do not start with index 0, there is a difference between calling and getting item, see `__call__()` for more details

      sage: sfibo = oeis('A039834')               # optional -- internet
      sage: sfibo.first_terms()[:10]              # optional -- internet
      (1, 1, 0, 1, -1, 2, -3, 5, -8, 13)

      sage: sfibo(-2)                             # optional -- internet
      1
      sage: sfibo(3)                              # optional -- internet
      2
      sage: sfibo.offsets()                       # optional -- internet
      (-2, 6)

      sage: sfibo[0]                              # optional -- internet
      1
      sage: sfibo[6]                              # optional -- internet
      -3

  `__call__`(k)

    Returns the element of the sequence `self` whith index `k`.

    INPUT:

    - `k` - integer.

    OUTPUT:

    - integer.

    Note

    The first index of the sequence `self` is not necessarily zero, it depends on the first offset of `self`. If the sequence represents the decimal expansion of a real number, the index 0 corresponds to the digit right after the decimal point.

    EXAMPLES:

        sage: f = oeis(45)                          # optional -- internet
        sage: f.first_terms()[:10]                  # optional -- internet
        (0, 1, 1, 2, 3, 5, 8, 13, 21, 34)

        sage: f(4)                                  # optional -- internet
        3

        sage: sfibo = oeis('A039834')               # optional -- internet
        sage: sfibo.first_terms()[:10]              # optional -- internet
        (1, 1, 0, 1, -1, 2, -3, 5, -8, 13)

        sage: sfibo(-2)                             # optional -- internet
        1
        sage: sfibo(4)                              # optional -- internet
        -3
        sage: sfibo.offsets()                       # optional -- internet
        (-2, 6)

    TESTS:

        sage: s = oeis._imaginary_sequence()
        sage: s(38)
        1
        sage: s(42)
        -1
        sage: s(2)
        Traceback (most recent call last):
        ...
        ValueError: Sequence A999999 is not defined (or known) for index 2

  `author`()

    Returns the author of the sequence in the encyclopedia.

    OUTPUT:

    - string.

    EXAMPLES:

        sage: f = oeis(45) ; f                      # optional -- internet
        A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

        sage: f.author()                            # optional -- internet
        '_N. J. A. Sloane_.'

    TESTS:

        sage: s = oeis._imaginary_sequence()
        sage: s.author()
        'Anonymous.'

  `browse`()

    Open the OEIS web page associated to the sequence `self` in a browser.

    EXAMPLES:

        sage: f = oeis(45) ; f                      # optional -- internet, webbrowser
        A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

        sage: f.browse()                            # optional -- internet, webbrowser

    TESTS:

        sage: s = oeis._imaginary_sequence()        # optional -- webbrowser
        sage: s.browse()                            # optional -- webbrowser

  `comments`()

    Return a tuple of comments associated to the sequence `self`.

    OUTPUT:

    - tuple of strings (with fancy formatting).

    EXAMPLES:

        sage: f = oeis(45) ; f                      # optional -- internet
        A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

        sage: f.comments()[:3]                      # optional -- internet
        ("Also called Lam{\\'e}'s sequence.",
         "F(n+2) = number of binary sequences of length n that have no consecutive 0's.",
         'F(n+2) = number of subsets of {1,2,...,n} that contain no consecutive integers.')

    TESTS:

        sage: s = oeis._imaginary_sequence()
        sage: s.comments()
        0: 42 is the product of the first 4 prime numbers, except 5 and perhaps 1.
        1: Apart from that, i have no comment.

  `cross_references`(fetch=False)

    Return a tuple of cross references associated to the sequence `self`.

    INPUT:

    - `fetch` - boolean (default: `False`).

    OUTPUT:

    - if `fetch` is `False`, return a list of OEIS IDs (strings).
    - if `fetch` if `True`, return a tuple of OEIS sequences.

    EXAMPLES:

        sage: nbalanced = oeis("A005598") ; nbalanced   # optional -- internet
        A005598: a(n)=1+sum((n-i+1)*phi(i),i=1..n).

        sage: nbalanced.cross_references()              # optional -- internet
        ('A049703', 'A049695', 'A103116', 'A000010')

        sage: nbalanced.cross_references(fetch=True)    # optional -- internet
        0: A049703: a(0) = 0; for n>0, a(n) = A005598(n)/2.
        1: A049695: Array T read by diagonals; T(i,j)=number of nonnegative slopes of lines determined by 2 lattice points in [ 0,i ] X [ 0,j ] if i>0; T(0,j)=1 if j>0; T(0,0)=0.
        2: A103116: A005598(n) - 1.
        3: A000010: Euler totient function phi(n): count numbers <= n and prime to n.

        sage: phi = _[3]                                # optional -- internet

    TESTS:

        sage: s = oeis._imaginary_sequence()
        sage: s.cross_references()
        ('A000042', 'A000024')

  `examples`()

    Return a tuple of examples associated to the sequence `self`.

    OUTPUT:

    - tuple of strings (with fancy formatting).

    EXAMPLES:

        sage: c = oeis(1203) ; c                    # optional -- internet
        A001203: Continued fraction expansion of Pi.

        sage: c.examples()                          # optional -- internet
        0: Pi = 3.1415926535897932384...
        1:    = 3 + 1/(7 + 1/(15 + 1/(1 + 1/(292 + ...))))
        2:    = [a_0; a_1, a_2, a_3, ...] = [3; 7, 15, 292, ...]

    TESTS:

        sage: s = oeis._imaginary_sequence()
        sage: s.examples()
        0: s(42) + s(43) = 0.

  `extensions_or_errors`()

    Return a tuple of extensions or errors associated to the sequence `self`.

    OUTPUT:

    - tuple of strings (with fancy formatting).

    EXAMPLES:

        sage: sfibo = oeis('A039834') ; sfibo       # optional -- internet
        A039834: a(n+2)=-a(n+1)+a(n) (signed Fibonacci numbers); or Fibonacci numbers (A000045) extended to negative indices.

        sage: sfibo.extensions_or_errors()[0]       # optional -- internet
        'Signs corrected by Len Smiley (smiley(AT)math.uaa.alaska.edu) and _N. J. A. Sloane_.'

    TESTS:

        sage: s = oeis._imaginary_sequence()
        sage: s.extensions_or_errors()
        0: This sequence does not contain errors.

  `first_terms`(number=None, absolute_value=False)

    INPUT:

    - `number` - (integer or `None`, default: `None`) the number of terms returned (if less than the number of available terms). When set to None, returns all the known terms.
    - `absolute_value` - (bool, default: `False`) when a sequence has negative entries, OEIS also stores the absolute values of its first terms, when `absolute_value` is set to `True`, you will get them.

    OUTPUT:

    - tuple of integers.

    EXAMPLES:

        sage: f = oeis(45) ; f                      # optional -- internet
        A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

        sage: f.first_terms()[:10]                  # optional -- internet
        (0, 1, 1, 2, 3, 5, 8, 13, 21, 34)

    TESTS:

        sage: s = oeis._imaginary_sequence()
        sage: s.first_terms()
        (1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)
        sage: s.first_terms(5)
        (1, 1, 1, 1, -1)
        sage: s.first_terms(5, absolute_value=True)
        (1, 1, 1, 1, 1)

        sage: s = oeis._imaginary_sequence(keywords='full')
        sage: s(40)
        Traceback (most recent call last):
        ...
        TypeError: You found a sign inconsistency, please contact OEIS

        sage: s = oeis._imaginary_sequence(keywords='sign,full')
        sage: s(40)
        1

        sage: s = oeis._imaginary_sequence(keywords='nonn,full')
        sage: s(42)
        1

  `formulas`()

    Return a tuple of formulas associated to the sequence `self`.

    OUTPUT:

    - tuple of strings (with fancy formatting).

    EXAMPLES:

        sage: f = oeis(45) ; f                      # optional -- internet
        A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

        sage: f.formulas()[1]                       # optional -- internet
        'F(n) = ((1+sqrt(5))^n-(1-sqrt(5))^n)/(2^n*sqrt(5)).'

    TESTS:

        sage: s = oeis._imaginary_sequence()
        sage: s.formulas()
        0: For n big enough, s(n+1) - s(n) = 0.

  `id`(format='A')

    The ID of the sequence `self` is the A-number that identifies `self`.

    INPUT:

    - `format` - (string, default: ‘A’).

    OUTPUT:

    - if `format` is set to ‘A’, returns a string of the form ‘A000123’.
    - if `format` is set to ‘int’ returns an integer of the form 123.

    EXAMPLES:

        sage: f = oeis(45) ; f                      # optional -- internet
        A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

        sage: f.id()                                # optional -- internet
        'A000045'

        sage: f.id(format='int')                    # optional -- internet
        45

    TESTS:

        sage: s = oeis._imaginary_sequence()
        sage: s.id()
        'A999999'
        sage: s.id(format='int')
        999999

  `is_finite`()

    Tells whether the sequence is finite.

    Currently, OEIS only provides a keyword when the sequence is known to be finite. So, when this keyword is not there, we do not know whether it is infinite or not.

    OUTPUT:

    - Returns `True` when the sequence is known to be finite.
    - Returns `Unknown` otherwise.

    Todo

    Ask OEIS for a keyword ensuring that a sequence is infinite.

    EXAMPLES:

        sage: s = oeis('A114288') ; s               # optional -- internet
        A114288: Lexicographically minimal solution of any 9 X 9 sudoku, read by rows.

        sage: s.is_finite()                         # optional -- internet
        True

        sage: f = oeis(45) ; f                      # optional -- internet
        A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

        sage: f.is_finite()                         # optional -- internet
        Unknown

    TESTS:

        sage: s = oeis._imaginary_sequence()
        sage: s.is_finite()
        Unknown

        sage: s = oeis._imaginary_sequence('nonn,finit')
        sage: s.is_finite()
        True

  `is_full`()

    Tells whether the sequence `self` is full, that is, if all its elements are listed in `self.first_terms()`.

    Currently, OEIS only provides a keyword when the sequence is known to be full. So, when this keyword is not there, we do not know whether some elements are missing or not.

    OUTPUT:

    - Returns `True` when the sequence is known to be full.
    - Returns `Unknown` otherwise.

    EXAMPLES:

        sage: s = oeis('A114288') ; s               # optional -- internet
        A114288: Lexicographically minimal solution of any 9 X 9 sudoku, read by rows.

        sage: s.is_full()                           # optional -- internet
        True

        sage: f = oeis(45) ; f                      # optional -- internet
        A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

        sage: f.is_full()                           # optional -- internet
        Unknown

    TESTS:

        sage: s = oeis._imaginary_sequence()
        sage: s.is_full()
        Unknown

        sage: s = oeis._imaginary_sequence('nonn,full,finit')
        sage: s.is_full()
        True

  `keywords`()

    Return the keywords associated to the sequence `self`.

    OUTPUT:

    - tuple of strings.

    EXAMPLES:

        sage: f = oeis(45) ; f                      # optional -- internet
        A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

        sage: f.keywords()                          # optional -- internet
        ('core', 'nonn', 'easy', 'nice', 'changed')

    TESTS:

        sage: s = oeis._imaginary_sequence()
        sage: s.keywords()
        ('sign', 'easy')

        sage: s = oeis._imaginary_sequence(keywords='nonn,hard')
        sage: s.keywords()
        ('nonn', 'hard')

  `links`(browse=None, format='guess')

    Return, display or browse links associated to the sequence `self`.

    INPUT:

    - `browse` - an integer, a list of integers, or the word ‘all’ (default: `None`) : which links to open in a web browser.
    - `format` - string (default: ‘guess’) : how to display the links.

    OUTPUT:

    - tuple of strings (with fancy formatting):
        - if `format` is `url`, returns a tuple of absolute links without description.
        - if `format` is `html`, returns nothing but prints a tuple of clickable absolute links in their context.
        - if `format` is `guess`, adapts the output to the context (command line or notebook).
        - if `format` is `raw`, the links as they appear in the database, relative links are not made absolute.

    EXAMPLES:

        sage: f = oeis(45) ; f                      # optional -- internet
        A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

        sage: f.links(format='url')                             # optional -- internet
        0: http://oeis.org/A000045/b000045.txt
        1: http://www.schoolnet.ca/vp-pv/amof/e_fiboI.htm
        ...

        sage: f.links(format='raw')                 # optional -- internet
        0: N. J. A. Sloane, <a href="/A000045/b000045.txt">The first 2000 Fibonacci numbers: Table of n, F(n) for n = 0..2000</a>
        1: Amazing Mathematical Object Factory, <a href="http://www.schoolnet.ca/vp-pv/amof/e_fiboI.htm">Information on the Fibonacci sequences</a>
        ...

    TESTS:

        sage: s = oeis._imaginary_sequence()
        sage: s.links(format='raw')[2]
        'Do not confuse with the sequence <a href="/A000042">A000042</a> or the sequence <a href="/A000024">A000024</a>'

        sage: s.links(format='url')[3]
        'http://oeis.org/A000024'

        sage: s.links(format="html")
        <html><font color='black'>0: Wikipedia, <a href="http://en.wikipedia.org/wiki/42_(number)">42 (number)</a>
        1: See. also <a href="http://trac.sagemath.org/sage_trac/ticket/42">trac ticket #42</a>
        ...

  `name`()

    Return the name of the sequence `self`.

    OUTPUT:

    - string.

    EXAMPLES:

        sage: f = oeis(45) ; f                      # optional -- internet
        A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

        sage: f.name()                              # optional -- internet
        'Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.'

    TESTS:

        sage: s = oeis._imaginary_sequence()
        sage: s.name()
        'The opposite of twice the characteristic sequence of 42 plus one, starting from 38.'

  `natural_object`()

    Return the natural object associated to the sequence `self`.

    OUTPUT:

    - If the sequence `self` corresponds to the digits of a real

        number, returns the associated real number (as an element of RealLazyField()).

    - If the sequence `self` corresponds to the convergents of a

        continued fraction, returns the associated continued fraction (as an element of ContinuedFractionField()).

    Warning

    This method forgets the fact that the returned sequence may not be complete.

    Todo

    - ask OEIS to add a keyword telling whether the sequence comes from a power series, e.g. for http://oeis.org/A000182
    - discover other possible conversions.

    EXAMPLES:

        sage: g = oeis("A002852") ; g               # optional -- internet
        A002852: Continued fraction for Euler's constant (or Euler-Mascheroni constant) gamma.

        sage: x = g.natural_object() ; x.parent()   # optional -- internet
        Field of all continued fractions

        sage: x[:20] == continued_fraction(euler_gamma, nterms=20)  # optional -- internet
        True

        sage: ee = oeis('A001113') ; ee             # optional -- internet
        A001113: Decimal expansion of e.

        sage: x = ee.natural_object() ; x           # optional -- internet
        2.718281828459046?

        sage: x.parent()                            # optional -- internet
        Real Lazy Field

        sage: x == RR(e)                            # optional -- internet
        True

        sage: av = oeis('A087778') ; av             # optional -- internet
        A087778: Decimal expansion of Avogadro's constant.

        sage: av.natural_object()                   # optional -- internet
        6.022141000000000?e23

        sage: fib = oeis('A000045') ; fib           # optional -- internet
        A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

        sage: x = fib.natural_object() ; x.parent()         # optional -- internet
        Category of sequences in Non negative integer semiring

        sage: sfib = oeis('A039834') ; sfib         # optional -- internet
        A039834: a(n+2)=-a(n+1)+a(n) (signed Fibonacci numbers); or Fibonacci numbers (A000045) extended to negative indices.

        sage: x = sfib.natural_object() ; x.parent()    # optional -- internet
        Category of sequences in Integer Ring

    TESTS:

        sage: s = oeis._imaginary_sequence('nonn,cofr')
        sage: s.natural_object().parent()
        Field of all continued fractions

        sage: s = oeis._imaginary_sequence('nonn')
        sage: s.natural_object().parent()
        Category of sequences in Non negative integer semiring

        sage: s = oeis._imaginary_sequence()
        sage: s.natural_object().parent()
        Category of sequences in Integer Ring

  `offsets`()

    Return the offsets of the sequence `self`.

    The first offset is the subscript of the first term in the sequence `self`. When, the sequence represents the decimal expansion of a real number, it corresponds to the number of digits of its integer part.

    The second offset is the first term in the sequence `self` (starting from 1) whose absolute value is greater than 1. This is set to 1 if all the terms are 0 or +-1.

    OUTPUT:

    - tuple of two elements.

    EXAMPLES:

        sage: f = oeis(45) ; f                      # optional -- internet
        A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

        sage: f.offsets()                           # optional -- internet
        (0, 4)

        sage: f.first_terms()[:4]                   # optional -- internet
        (0, 1, 1, 2)

    TESTS:

        sage: s = oeis._imaginary_sequence()
        sage: s.offsets()
        (38, 4)

  `old_IDs`()

    Returns the IDs of the sequence `self` corresponding to ancestors of OEIS.

    OUTPUT:

    - a tuple of at most two strings. When the string starts with \(M\), it corresponds to the ID of “The Encyclopedia of Integer Sequences” of 1995. When the string starts with \(N\), it corresponds to the ID of the “Handbook of Integer Sequences” of 1973.

    EXAMPLES:

        sage: f = oeis(45) ; f                      # optional -- internet
        A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

        sage: f.old_IDs()                           # optional -- internet
        ('M0692', 'N0256')

    TESTS:

        sage: s = oeis._imaginary_sequence()
        sage: s.old_IDs()
        ('M9999', 'N9999')

  `programs`(language='other')

    Returns programs implementing the sequence `self` in the given `language`.

    INPUT:

    - `language` - string (default: ‘other’) - the language of the program. Current values are: ‘maple’, ‘mathematica’ and ‘other’.

    OUTPUT:

    - tuple of strings (with fancy formatting).

    Todo

    ask OEIS to add a “Sage program” field in the database ;)

    EXAMPLES:

        sage: ee = oeis('A001113') ; ee             # optional -- internet
        A001113: Decimal expansion of e.

        sage: ee.programs()[0]                      # optional -- internet
        '(PARI) { default(realprecision, 50080); x=exp(1); for (n=1, 50000, d=floor(x); x=(x-d)*10; write("b001113.txt", n, " ", d)); } [From Harry J. Smith, Apr 15 2009]'

    TESTS:

        sage: s = oeis._imaginary_sequence()
        sage: s.programs()
        0: (Python)
        1: def A999999(n):
        2:     assert(isinstance(n, (int, Integer))), "n must be an integer."
        3:     if n < 38:
        4:         raise ValueError("The value %s is not accepted." %str(n)))
        5:     elif n == 42:
        6:         return -1
        7:     else:
        8:         return 1

        sage: s.programs('maple')
        0: Do not even try, Maple is not able to produce such a sequence.

        sage: s.programs('mathematica')
        0: Mathematica neither.

  `raw_entry`()

    Return the raw entry of the sequence `self`, in the OEIS format.

    OUTPUT:

    - string.

    EXAMPLES:

        sage: f = oeis(45) ; f                      # optional -- internet
        A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

        sage: print f.raw_entry()                   # optional -- internet
        %I A000045 M0692 N0256
        %S A000045 0,1,1,2,3,5,8,13,21,34,55,89,144,...
        %T A000045 10946,17711,28657,46368,...
        ...

    TESTS:

        sage: s = oeis._imaginary_sequence()
        sage: s.raw_entry() == oeis._imaginary_entry('sign,easy')
        True

  `references`()

    Return a tuple of references associated to the sequence `self`.

    OUTPUT:

    - tuple of strings (with fancy formatting).

    EXAMPLES:

        sage: w = oeis(7540) ; w                    # optional -- internet
        A007540: Wilson primes: primes p such that (p-1)! == -1 (mod p^2).

        sage: w.references()                        # optional -- internet
        0: A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 52.
        1: C. Clawson, Mathematical Mysteries, Plenum Press, 1996, p. 180.
        2: Edgar Costa, Robert Gerbicz, and David Harvey, <a href="http://arxiv.org/abs/1209.3436">A search for Wilson primes</a>, 2012
        ...

        sage: _[0]                                  # optional -- internet
        'A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 52.'

    TESTS:

        sage: s = oeis._imaginary_sequence()
        sage: s.references()[1]
        'Lewis Carroll, The Hunting of the Snark.'

  `show`()

    Display most available informations about the sequence `self`.

    EXAMPLES:

        sage: s = oeis(12345)                       # optional -- internet
        sage: s.show()                              # optional -- internet
        ID
        A012345
        <BLANKLINE>
        NAME
        sinh(arcsin(x)*arcsin(x))=2/2!*x^2+8/4!*x^4+248/6!*x^6+11328/8!*x^8...
        <BLANKLINE>
        FIRST TERMS
        (2, 8, 248, 11328, 849312, 94857600, 14819214720, 3091936512000, ...
        <BLANKLINE>
        KEYWORDS
        ('nonn',)
        <BLANKLINE>
        OFFSETS
        (0, 1)
        URL
        http://oeis.org/A012345
        <BLANKLINE>
        AUTHOR
        Patrick Demichel (patrick.demichel(AT)hp.com)
        <BLANKLINE>

    TESTS:

        sage: s = oeis._imaginary_sequence()
        sage: s.show()
        ID
        A999999
        <BLANKLINE>
        NAME
        The opposite of twice the characteristic sequence of 42 plus ...
        FIRST TERMS
        (1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
        <BLANKLINE>
        COMMENTS
        0: 42 is the product of the first 4 prime numbers, except ...
        1: Apart from that, i have no comment.
        ...

  `url`()

    Return the URL of the page associated to the sequence `self`.

    OUTPUT:

    - string.

    EXAMPLES:

        sage: f = oeis(45) ; f                      # optional -- internet
        A000045: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

        sage: f.url()                               # optional -- internet
        'http://oeis.org/A000045'

    TESTS:

        sage: s = oeis._imaginary_sequence()
        sage: s.url()
        'http://oeis.org/A999999'

`sage.databases.oeis.``to_tuple`(string)

  x.__init__(...) initializes x; see help(type(x)) for signature

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