Coverage for local/lib/python2.7/site-packages/sage/databases/oeis.py: 74%

2: 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.

3: A246674: Run Length Transform of A000225.

sage: [u.id() for u in search] # optional -- internet

['A001203', 'A082495', 'A165416', 'A246674']

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, 1, 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() ; type(x) # optional -- internet

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

sage: len(p.first_terms()) # optional -- internet

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 topologies on n labeled points satisfying axioms T_0-T_4.

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).coefficients()] ; 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: number of ways to partition a set of n labeled elements.

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

sage: b.formulas()[0] # optional -- internet

'E.g.f.: exp(exp(x) - 1).'

sage: [i for i in b.comments() if 'prime' in i][-1] # optional -- internet

'Number n is prime if mod(a(n)-2,n) = 0. -_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]

.. SEEALSO::

- If you plan to do a lot of automatic searches for subsequences, you

should consider installing :mod:`SloaneEncyclopedia

<sage.databases.sloane>`, a local partial copy of the OEIS.

- Some infinite OEIS sequences are implemented in Sage, via the

:mod:`sloane_functions <sage.combinat.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

-------------------

"""

#*****************************************************************************

# Distributed under the terms of the GNU General Public License (GPL)

# as published by the Free Software Foundation; either version 2 of

# the License, or (at your option) any later version.

# http://www.gnu.org/licenses/

#*****************************************************************************

from __future__ import print_function

from six.moves.urllib.request import urlopen

from six.moves.urllib.parse import urlencode

from sage.structure.sage_object import SageObject

from sage.rings.integer import Integer

from sage.misc.misc import verbose

from sage.misc.cachefunc import cached_method

from sage.misc.flatten import flatten

from sage.misc.unknown import Unknown

from sage.misc.misc import embedded

from sage.misc.html import HtmlFragment

from collections import defaultdict

import re

oeis_url = 'http://oeis.org/'

def _fetch(url):

r"""

Fetch the given ``url``.

INPUT:

- ``url`` - a string corresponding to the URL to be fetched.

OUTPUT:

- a string representing the fetched web page.

TESTS::

sage: from sage.databases.oeis import _fetch, oeis_url

sage: _fetch(oeis_url + 'hints.html')[-8:-1] # optional -- internet

'</html>'

"""

try:

verbose("Fetching URL %s ..." % url, caller_name='OEIS')

f = urlopen(url)

result = f.read()

f.close()

return result

except IOError as msg:

raise IOError("%s\nError fetching %s." % (msg, url))

def _urls(html_string):

r"""

Return the list of URLs contained in ``html_string``.

Only URLs provided by HTML hyperlinks (``href`` attribute of ``<a>`` tags)

in are returned, not text strings starting with ``http://``.

INPUT:

- ``html_string`` - a string representing some HTML code.

OUTPUT:

- a list of (string) URLs contained in ``html_string``.

EXAMPLES::

sage: from sage.databases.oeis import _urls

sage: html = 'http://example.com is not a link, but <a href="http://sagemath.org/">sagemath</a> is'

sage: _urls(html)

['http://sagemath.org/']

"""

urls = []

from HTMLParser import HTMLParser

class MyHTMLParser(HTMLParser):

def handle_starttag(self, tag, attrs):

if tag == 'a':

for attr in attrs:

if attr[0] == 'href':

urls.append(attr[1])

MyHTMLParser().feed(html_string)

return urls

to_tuple = lambda string: tuple(Integer(x) for x in string.split(",") if x)

class OEIS:

r"""

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; T(n,1) = 1 for n >= 1; T(n,k) = T(n-1,k-2) + T(n-1,k-1) for k = 2..2n-1, n >= 2.

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()[4] # 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; T(n,1) = 1 for n >= 1; T(n,k) = T(n-1,k-2) + T(n-1,k-1) for k = 2..2n-1, n >= 2.

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; T(n,1) = 1 for n >= 1; T(n,k) = T(n-1,k-2) + T(n-1,k-1) for k = 2..2n-1, n >= 2.

2: A001129: Iccanobif numbers: reverse digits of two previous terms and add.

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

"""

def __call__(self, query, max_results=3, first_result=0):

r"""

See the documentation of :class:`OEIS`.

TESTS::

sage: oeis()

Traceback (most recent call last):

...

TypeError: __call__() takes at least 2 arguments (1 given)

"""

if isinstance(query, str):

if re.match('^A[0-9]{6}$', query):

return self.find_by_id(query)

else:

return self.find_by_description(query, max_results, first_result)

elif isinstance(query, (int, Integer)):

return self.find_by_id(query)

elif isinstance(query, (list, tuple)):

return self.find_by_subsequence(query, max_results, first_result)

def __repr__(self):

r"""

Return the representation of ``self``.

TESTS::

sage: oeis

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

"""

return "The On-Line Encyclopedia of Integer Sequences (%s)" % oeis_url

def find_by_id(self, ident):

r"""

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.

"""

if not isinstance(ident, str):

ident = str(ident)

ident = 'A000000'[:-len(ident)] + ident

options = {'q': ident, 'n': '1', 'fmt': 'text'}

url = oeis_url + "search?" + urlencode(options)

sequence = _fetch(url).split('\n\n')[2]

return OEISSequence(sequence)

def find_by_description(self, description, max_results=3, first_result=0):

r"""

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}, ...

"""

options = {'q': description,

'n': str(max_results),

'fmt': 'text',

'start': str(first_result)}

url = oeis_url + "search?" + urlencode(options)

sequence_list = _fetch(url).split('\n\n')[2:-1]

return FancyTuple([OEISSequence(_) for _ in sequence_list])

def find_by_subsequence(self, subsequence, max_results=3, first_result=0):

r"""

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: A212804: Expansion of (1-x)/(1-x-x^2).

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.

"""

subsequence = str(subsequence)[1:-1]

return self.find_by_description(subsequence, max_results, first_result)

def browse(self):

r"""

Open the OEIS web page in a browser.

EXAMPLES::

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

"""

import webbrowser

webbrowser.open(oeis_url)

def _imaginary_entry(self, keywords=''):

r"""

This is an imaginary entry of an OEIS sequence for offline tests.

INPUT:

- ``keywords`` - a string corresponding to the keyword field of the

sequence.

OUTPUT:

- a string representing the entry of the sequence.

TESTS::

sage: oeis._imaginary_entry().split('\n')[0]

'%I A999999 M9999 N9999'

sage: from sage.databases.oeis import OEISSequence

sage: keywords = 'simon,cussonet'

sage: s = OEISSequence(oeis._imaginary_entry(keywords))

sage: ','.join(s.keywords()) == keywords

True

"""

return ('%I A999999 M9999 N9999\n'

'%S A999999 1,1,1,1,1,1,1,1,\n'

'%T A999999 1,1,1,1,1,1,1,1,1,\n'

'%U A999999 1,1,1,1,1,1,1,1,1\n'

'%V A999999 1,1,1,1,-1,1,1,1,\n'

'%W A999999 1,1,1,1,1,1,1,1,1,\n'

'%X A999999 1,1,1,1,1,1,1,1,1\n'

'%N A999999 The opposite of twice the characteristic sequence of 42 plus one, starting from 38.\n'

'%D A999999 Lewis Carroll, Alice\'s Adventures in Wonderland.\n'

'%D A999999 Lewis Carroll, The Hunting of the Snark.\n'

'%D A999999 Deep Thought, The Answer to the Ultimate Question of Life, The Universe, and Everything.\n'

'%H A999999 Wikipedia, <a href="https://en.wikipedia.org/wiki/42_(number)">42 (number)</a>\n'

'%H A999999 See. also <a href="https://trac.sagemath.org/sage_trac/ticket/42">trac ticket #42</a>\n'

'%H A999999 Do not confuse with the sequence <a href="/A000042">A000042</a> or the sequence <a href="/A000024">A000024</a>\n'

'%H A999999 The string http://42.com is not a link.\n'

'%F A999999 For n big enough, s(n+1) - s(n) = 0.\n'

'%Y A999999 Related sequences are A000042 and its friend A000024.\n'

'%A A999999 Anonymous.\n'

'%O A999999 38,4\n'

'%E A999999 This sequence does not contain errors.\n'

'%e A999999 s(42) + s(43) = 0.\n'

'%p A999999 Do not even try, Maple is not able to produce such a sequence.\n'

'%t A999999 Mathematica neither.\n'

'%o A999999 (Python)\n'

'%o A999999 def A999999(n):\n'

'%o A999999 assert(isinstance(n, (int, Integer))), "n must be an integer."\n'

'%o A999999 if n < 38:\n'

'%o A999999 raise ValueError("The value %s is not accepted." %str(n)))\n'

'%o A999999 elif n == 42:\n'

'%o A999999 return -1\n'

'%o A999999 else:\n'

'%o A999999 return 1\n'

'%K A999999 ' + keywords + '\n'

'%C A999999 42 is the product of the first 4 prime numbers, except 5 and perhaps 1.\n'

'%C A999999 Apart from that, i have no comment.')

def _imaginary_sequence(self, keywords='sign,easy'):

r"""

This is the OEIS sequence corresponding to the imaginary entry.

Its main purpose is to allow offline doctesting.

INPUT:

- ``keywords`` - string (default: 'sign,easy'), a list of words

separated by commas.

OUTPUT:

- OEIS sequence.

TESTS::

sage: s = oeis._imaginary_sequence()

sage: s

A999999: The opposite of twice the characteristic sequence of 42 plus one, starting from 38.

sage: s[4]

-1

sage: s(42)

-1

"""

return OEISSequence(self._imaginary_entry(keywords))

class OEISSequence(SageObject):

r"""

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

:class:`OEIS`.

.. NOTE::

Since some sequences do not start with index 0, there is a difference

between calling and getting item, see :meth:`__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

sage: sfibo(3) # optional -- internet

sage: sfibo.offsets() # optional -- internet

(-2, 6)

sage: sfibo[0] # optional -- internet

sage: sfibo[6] # optional -- internet

-3

.. automethod:: __call__

"""

def __init__(self, entry):

r"""

Initializes an OEIS sequence.

TESTS::

sage: sfibo = oeis('A039834') # optional -- internet

Handle dead sequences: see :trac:`17330` ::

sage: oeis(17) # optional -- internet

.. RuntimeWarning: This sequence is dead "A000017: Erroneous version of A032522."

A000017: Erroneous version of A032522.

sage: s = oeis._imaginary_sequence()

"""

self._raw = entry

self._id = entry[3:10]

self._fields = defaultdict(list)

for line in entry.splitlines():

self._fields[line[1]].append(line[11:])

if 'dead' in self.keywords():

("This sequence is dead: \""+self.name()+"\"")

from warnings import warn

warn('This sequence is dead "'+self.id()+": "+self.name()+'"', RuntimeWarning)

def id(self, format='A'):

r"""

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

TESTS::

sage: s = oeis._imaginary_sequence()

sage: s.id()

'A999999'

sage: s.id(format='int')

999999

"""

if format == 'A':

return self._id

elif format == 'int':

return Integer(self._id[1:].lstrip("0"))

def raw_entry(self):

r"""

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

"""

return self._raw

def name(self):

r"""

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.'

"""

return self._fields['N'][0]

def old_IDs(self):

r"""

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')

"""

return tuple(self._fields['I'][0].split(' '))

def offsets(self):

r"""

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)

"""

return to_tuple(self._fields['O'][0])

def author(self):

r"""

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_, Apr 30 1991'

TESTS::

sage: s = oeis._imaginary_sequence()

sage: s.author()

'Anonymous.'

"""

return self._fields['A'][0]

def keywords(self):

r"""

Return the keywords associated to the sequence ``self``.

OUTPUT:

- tuple of strings.

EXAMPLES::

sage: f = oeis(53) ; f # optional -- internet

A000053: Local stops on New York City Broadway line (IRT #1) subway.

sage: f.keywords() # optional -- internet

('nonn', 'fini', 'full')

TESTS::

sage: s = oeis._imaginary_sequence()

sage: s.keywords()

('sign', 'easy')

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

sage: s.keywords()

('nonn', 'hard')

"""

return tuple(self._fields['K'][0].split(','))

def natural_object(self):

r"""

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.

.. 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() ; type(x) # optional -- internet

sage: RDF(x) == RDF(euler_gamma) # optional -- internet

True

sage: cfg = continued_fraction(euler_gamma)

sage: x[:90] == cfg[:90] # 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.universe() # optional -- internet

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.universe() # optional -- internet

Integer Ring

TESTS::

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

sage: type(s.natural_object())

sage: s = oeis._imaginary_sequence('nonn')

sage: s.natural_object().universe()

Non negative integer semiring

sage: s = oeis._imaginary_sequence()

sage: s.natural_object().universe()

Integer Ring

"""

if 'cofr' in self.keywords() and not 'frac' in self.keywords():

from sage.rings.continued_fraction import continued_fraction

return continued_fraction(self.first_terms())

elif 'cons' in self.keywords():

offset = self.offsets()[0]

terms = self.first_terms() + tuple([0] * abs(offset))

from sage.rings.real_lazy import RealLazyField

return RealLazyField()('0' + ''.join(map(str, terms[:offset])) + '.' + ''.join(map(str, terms[offset:])))

elif 'nonn' in self.keywords():

from sage.structure.sequence import Sequence

from sage.rings.semirings.non_negative_integer_semiring import NN

return Sequence(self.first_terms(), NN)

else:

from sage.structure.sequence import Sequence

from sage.rings.integer_ring import ZZ

return Sequence(self.first_terms(), ZZ)

def is_finite(self):

r"""

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 earliest 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

"""

if 'finit' in self.keywords() or 'full' in self.keywords():

return True

else:

return Unknown

def is_full(self):

r"""

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 earliest 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

"""

if 'full' in self.keywords():

return True

else:

return Unknown

@cached_method

def first_terms(self, number=None, absolute_value=False):

r"""

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)

Handle dead sequences: see :trac:`17330` ::

sage: oeis(17).first_terms(12) # optional -- internet

oeis(17).first_terms(12)

.. RuntimeWarning: This sequence is dead "A000017: Erroneous version of A032522."

warn('This sequence is dead "'+self.id()+": "+self.name()+'"', RuntimeWarning)

(1, 0, 0, 2, 2, 4, 8, 4, 16, 12, 48, 80)

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)

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

sage: s(42)

"""

if absolute_value or ('nonn' in self.keywords()) or ('dead' in self.keywords()):

fields = ['S', 'T', 'U']

elif ('sign' in self.keywords()):

fields = ['V', 'W', 'X']

else:

raise TypeError("You found a sign inconsistency, please contact OEIS")

return to_tuple(" ".join(flatten([self._fields[a] for a in fields])))[:number]

def _repr_(self):

r"""

Prints the sequence number and a short summary of this sequence.

OUTPUT:

- string.

EXAMPLES::

sage: f = oeis(45) # optional -- internet

sage: f # optional -- internet

A000045: 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

A999999: The opposite of twice the characteristic sequence of 42 plus one, starting from 38.

"""

return "%s: %s" % (self.id(), self.name())

def __call__(self, k):

r"""

Return the element of the sequence ``self`` with 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

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

sage: sfibo(4) # optional -- internet

-3

sage: sfibo.offsets() # optional -- internet

(-2, 6)

TESTS::

sage: s = oeis._imaginary_sequence()

sage: s(38)

sage: s(42)

-1

sage: s(2)

Traceback (most recent call last):

...

ValueError: Sequence A999999 is not defined (or known) for index 2

"""

offset = self.offsets()[0]

if 'cons' in self.keywords():

offset = - offset

n = k - offset

if not 0 <= n < len(self.first_terms()):

raise ValueError("Sequence %s is not defined (or known) for index %s" % (self.id(), k))

return self.first_terms()[n]

def __getitem__(self, i):

r"""

Return the ``i``th element of sequence ``self``, viewed as a tuple.

The first element appearing in the sequence ``self``corresponds to

``self[0]``. Do not confuse with calling ``self(k)``.

INPUT:

- ``i`` - integer.

OUTPUT:

- integer.

EXAMPLES::

sage: sfibo = oeis('A039834') # optional -- internet

sage: sfibo[8] # optional -- internet

-8

sage: sfibo(8) # optional -- internet

-21

TESTS::

sage: s = oeis._imaginary_sequence()

sage: s[2]

sage: s[4]

-1

sage: s[38]

Traceback (most recent call last):

...

IndexError: tuple index out of range

"""

return self.first_terms()[i]

def __iter__(self):

r"""

Iterates over the first terms of ``self``, and raises an error if

those first terms are exhausted and the real associated sequence

still have terms to produce.

OUTPUT:

- integer.

EXAMPLES::

sage: p = oeis('A085823') ; p # optional -- internet

A085823: Numbers in which all substrings are primes.

sage: for i in p: # optional -- internet

....: print(i)

373

sage: w = oeis(7540) ; w # optional -- internet

A007540: Wilson primes: primes p such that (p-1)! == -1 (mod p^2).

sage: i = w.__iter__() # optional -- internet

sage: next(i) # optional -- internet

563

sage: next(i) # optional -- internet

Traceback (most recent call last):

...

LookupError: Future values not provided by OEIS.

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: for i in f: # optional -- internet

....: print(i)

Traceback (most recent call last):

...

LookupError: Future values not provided by OEIS.

TESTS::

sage: s = oeis._imaginary_sequence()

sage: for i in s:

....: pass

Traceback (most recent call last):

...

LookupError: Future values not provided by OEIS.

sage: for i in s:

....: if i == -1:

....: print(i)

....: break

-1

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

sage: for i in s: pass

"""

for x in self.first_terms():

yield x

if not self.is_full():

raise LookupError("Future values not provided by OEIS.")

def __eq__(self, other):

r"""

Returns ``True`` if ``self`` is equal to ``other`` and ``False``

otherwise. Two integer sequences are considered equal if they have the

same OEIS ID.

INPUT:

- ``other`` - an oeis sequence.

OUTPUT:

- boolean.

EXAMPLES::

sage: oeis([1,2,3,5,8,13])[0] == oeis(45) # optional -- internet

True

TESTS::

sage: s = oeis._imaginary_sequence()

sage: s == oeis._imaginary_sequence()

True

"""

return self.id() == other.id()

def __ne__(self, other):

r"""

Returns ``True`` if ``self`` has a different OEIS ID than ``other`` and

``False`` otherwise.

INPUT:

- ``other`` - an oeis sequence.

OUTPUT:

- boolean.

EXAMPLES::

sage: oeis([1,2,3,5,8,13])[0] != oeis(40) # optional -- internet

True

TESTS::

sage: s = oeis._imaginary_sequence()

sage: s != oeis._imaginary_sequence()

False

"""

return not self == other

def references(self):

r"""

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: R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 29.

3: G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 80.

...

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.'

"""

return FancyTuple(self._fields['D'])

def links(self, browse=None, format='guess'):

r"""

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: HTML = s.links(format="html"); HTML

0: Wikipedia, <a href="https://en.wikipedia.org/wiki/42_(number)">42 (number)</a>

1: See. also <a href="https://trac.sagemath.org/sage_trac/ticket/42">trac ticket #42</a>

...

sage: type(HTML)

"""

url_absolute = lambda s: re.sub('\"\/', '\"' + oeis_url, s)

if browse is None:

if format == 'guess':

if embedded():

return self.links(format='html')

else:

return self.links(format='url')

elif format == 'raw':

return FancyTuple(self._fields['H'])

elif format == 'html':

return HtmlFragment(FancyTuple([url_absolute(_) for _ in self._fields['H']]))

elif format == 'url':

url_list = flatten([_urls(url_absolute(string)) for string in self._fields['H']])

return FancyTuple(url_list)

else:

import webbrowser

url_list = flatten([_urls(url_absolute(string)) for string in self._fields['H']])

if isinstance(browse, (int, Integer)):

webbrowser.open(url_list[browse])

elif isinstance(browse, (list, tuple)):

for url_number in browse:

webbrowser.open(url_list[url_number])

elif browse == 'all':

for url in url_list:

webbrowser.open(url)

def formulas(self):

r"""

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()[2] # 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.

"""

return FancyTuple(self._fields['F'])

def cross_references(self, fetch=False):

r"""

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')

"""

ref_list = re.findall('A[0-9]{6}', " ".join(self._fields['Y']))

if fetch:

return FancyTuple([oeis.find_by_id(_) for _ in ref_list])

else:

return tuple(ref_list)

def extensions_or_errors(self):

r"""

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_ and _N. J. A. Sloane_.'

TESTS::

sage: s = oeis._imaginary_sequence()

sage: s.extensions_or_errors()

0: This sequence does not contain errors.

"""

return FancyTuple(self._fields['E'])

def examples(self):

r"""

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, 1, 292, ...]

TESTS::

sage: s = oeis._imaginary_sequence()

sage: s.examples()

0: s(42) + s(43) = 0.

"""

return FancyTuple(self._fields['e'])

def comments(self):

r"""

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

0: Also sometimes called Lamé's sequence.

1: F(n+2) = number of binary sequences of length n that have no consecutive 0's.

2: 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.

"""

return FancyTuple(self._fields['C'])

def url(self):

r"""

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'

"""

return oeis_url + self.id()

def browse(self):

r"""

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

"""

import webbrowser

webbrowser.open(self.url())

def show(self):

r"""

Display most available informations about the sequence ``self``.

EXAMPLES::

sage: s = oeis(12345) # optional -- internet

sage: s.show() # optional -- internet

A012345

NAME

Coefficients in the expansion sinh(arcsin(x)*arcsin(x)) = 2*x^2/2!+8*x^4/4!+248*x^6/6!+11328*x^8/8!+...

FIRST TERMS

(2, 8, 248, 11328, 849312, 94857600, 14819214720, 3091936512000, 831657655349760, 280473756197529600, 115967597965430077440, 57712257892456911912960, 34039765801079493369569280)

FORMULAS

...

OFFSETS

(0, 1)

URL

http://oeis.org/A012345

AUTHOR

Patrick Demichel (patrick.demichel(AT)hp.com)

TESTS::

sage: s = oeis._imaginary_sequence()

sage: s.show()

A999999

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, ...

COMMENTS

0: 42 is the product of the first 4 prime numbers, except ...

1: Apart from that, i have no comment.

...

"""

for s in ['id', 'name', 'first_terms', 'comments', 'references',

'links', 'formulas', 'examples', 'cross_references',

'programs', 'keywords', 'offsets', 'url', 'old_IDs',

'author', 'extensions_or_errors']:

if embedded() and s == 'links':

print(re.sub('_', ' ', s).upper())

getattr(self, s)()

print('\n')

else:

result = getattr(self, s)()

if result != '' and result != ('',) and result != ():

print(re.sub('_', ' ', s).upper())

print(str(result) + '\n')

def programs(self, language='other'):

r"""

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)); } \\\\ _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.

"""

if language == "maple":

return FancyTuple(self._fields['p'])

elif language == "mathematica":

return FancyTuple(self._fields['t'])

else:

return FancyTuple(self._fields['o'])

class FancyTuple(tuple):

r"""

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'

"""

def __repr__(self):

r"""

Prints the tuple with one value per line, each line begins with the

index of the value in ``self``.

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

"""

length = len(str(len(self) - 1))

return '\n'.join((('{0:>%d}' % length).format(str(i)) + ': ' + str(self[i]) for i in range(len(self))))

def __getslice__(self, i, j):

r"""

The slice of a FancyTuple remains a FancyTuple.

EXAMPLES::

sage: from sage.databases.oeis import FancyTuple

sage: t = FancyTuple(['zero', 'one', 'two', 'three', 4])

sage: t[-2:]

0: three

1: 4

TESTS::

sage: t = ('é', 'è', 'à', 'ç')

sage: t

('\xc3\xa9', '\xc3\xa8', '\xc3\xa0', '\xc3\xa7')

sage: FancyTuple(t)[2:4]

0: à

1: ç

"""

return FancyTuple(tuple(self).__getslice__(i, j))

oeis = OEIS()

Coverage for local/lib/python2.7/site-packages/sage/databases/oeis.py : 74%

219 statements 163 run 56 missing 0 excluded