Editing On-Line Encyclopedia of Integer Sequences

{{short description|Online database of integer sequences}}
{{Redirect|OEIS|the birth defect known as OEIS complex|Cloacal exstrophy}}
{{Infobox website
| name         = On-Line Encyclopedia of Integer Sequences
| logo         = [[File:OEIS banner.png|300px]]
| url          = {{url|https://oeis.org/}}
| commercial   = No<ref name="oeisfgoals">{{cite web |url=http://oeisf.org/index.html#GOALS |title=Goals of The OEIS Foundation Inc. |author=<!--Not stated--> |website=The OEIS Foundation Inc. |access-date=2017-11-06 |archive-url=https://web.archive.org/web/20131206172532/http://oeisf.org/index.html#GOALS |archive-date=2013-12-06 |url-status=dead }}</ref>
| registration = Optional<ref>Registration is required for editing entries or submitting new entries to the database</ref>
| launch_date  = {{start date and age|1996}}
| founded      = {{start date and age|1964}}
| author       = [[Neil Sloane]]
| chairman     = [[Neil Sloane]]
| president    = Russ Cox
| predecessor  = Handbook of Integer Sequences, [[Encyclopedia of Integer Sequences]]
| license = [[Creative Commons]] [[CC BY-SA]] 4.0<ref>{{Cite web |title=The OEIS End-User License Agreement - OeisWiki |url=https://oeis.org/wiki/The_OEIS_End-User_License_Agreement |access-date=2023-02-26 |website=oeis.org}}</ref>
}}

The '''On-Line Encyclopedia of Integer Sequences''' ('''OEIS''') is an online database of [[integer sequence]]s. It was created and maintained by [[Neil Sloane]] while researching at [[AT&T Labs]]. He transferred the [[intellectual property]] and hosting of the OEIS to the '''OEIS Foundation''' in 2009,<ref>{{Cite web |url=http://oeisf.org/index.html#IPXFER |title=Transfer of IP in OEIS to the OEIS Foundation Inc. |access-date=2010-06-01 |archive-url=https://web.archive.org/web/20131206172532/http://oeisf.org/index.html#IPXFER |archive-date=2013-12-06 |url-status=dead }}</ref> and is its chairman. 

OEIS records information on integer sequences of interest to both professional and [[List of amateur mathematicians|amateur]] [[mathematician]]s, and is widely cited. {{As of|2024|02|url=https://oeis.org/||df=UK}}, it contains over 370,000 sequences,<ref>{{Cite web |url=https://oeis.org|title=The On-Line Encyclopedia of Integer Sequences (OEIS)}}</ref> and is growing by approximately 30 entries per day.<ref>{{cite web |title=FAQ for the On-Line Encyclopedia of Integer Sequences |url=https://oeis.org/FAQ.html#Z27 |website=The On-Line Encyclopedia of Integer Sequences |publisher=OEIS Foundation |access-date=22 June 2024 }}</ref>

Each entry contains the leading terms of the sequence, [[Keyword (computer programming)|keyword]]s, mathematical motivations, literature links, and more, including the option to generate a [[Graph of a function|graph]] or play a [[Computer music|musical]] representation of the sequence. The database is [[Search engine (computing)|searchable]] by keyword, by [[subsequence]], or by any of 16 fields. There is also an advanced search function called SuperSeeker which runs a large number of different algorithms to identify sequences related to the input.<ref>{{Cite web |last=Sloane |first=Neil |date=2024 |title=The Email Servers and Superseeker |url=https://oeis.org/ol.html }}</ref>

==History==
[[File:Encyclopedia of Integer Sequences, 2nd edition, by N.J.A. Sloane.jpg|right|thumb|150px|Second edition of the book]]
[[Neil Sloane]] started collecting integer sequences as a graduate student in 1964 to support his work in [[combinatorics]].<ref>{{cite book | last=Borwein | first=Jonathan M. | title=Analytic Number Theory, Modular Forms and q-Hypergeometric Series  | chapter=Adventures with the OEIS | series=Springer Proceedings in Mathematics & Statistics | publisher=Springer International Publishing | publication-place=Cham | year=2017 | volume=221 | isbn=978-3-319-68375-1 | issn=2194-1009 | doi=10.1007/978-3-319-68376-8_9 | editor-first1 = George E. | editor-last1 = Andrews  | editor-first2 = Frank | editor-last2 = Garvan | pages = 123–138}}</ref><ref>{{cite news |first=James |last=Gleick |url=https://www.nytimes.com/1987/01/27/science/in-a-random-world-he-collects-patterns.html |title=In a 'random world,' he collects patterns |newspaper=The New York Times |date= January 27, 1987 |page=C1 }}</ref> The database was at first stored on [[punched card]]s. He published selections from the database in book form twice:
#'''''A Handbook of Integer Sequences''''' (1973, {{isbn|0-12-648550-X}}), containing 2,372 sequences in [[Lexicographical order|lexicographic order]] and assigned numbers from 1 to 2372.
#'''''The Encyclopedia of Integer Sequences''''' with [[Simon Plouffe]] (1995, {{isbn|0-12-558630-2}}), containing 5,488 sequences and assigned M-numbers from M0000 to M5487.  The Encyclopedia includes the references to the corresponding sequences (which may differ in their few initial terms) in ''A Handbook of Integer Sequences'' as N-numbers from N0001 to N2372 (instead of 1 to 2372.) The Encyclopedia includes the A-numbers that are used in the OEIS, whereas the Handbook did not.
[[File:OEIS-original web page.png|thumb|right|alt=1999 "Integer Sequences" web page|Sloane's "Integer Sequences" web page on the "AT&T research" web site as of 1999]]
These books were well-received and, especially after the second publication, mathematicians supplied Sloane with a steady flow of new sequences. The collection became unmanageable in book form, and when the database reached 16,000 entries Sloane decided to go online – first as an [[email]] service (August 1994), and soon thereafter as a website (1996). As a spin-off from the database work, Sloane founded the ''[[Journal of Integer Sequences]]'' in 1998.<ref>[http://www.cs.uwaterloo.ca/journals/JIS/ Journal of Integer Sequences]
 ({{ISSN|1530-7638}})</ref>
The database continues to grow at a rate of some 10,000 entries a year.
Sloane has personally managed 'his' sequences for almost 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the omnibus database.<ref>{{cite encyclopedia | url = http://oeis.org/wiki/Editorial_Board | title = Editorial Board | encyclopedia = On-Line Encyclopedia of Integer Sequences}}</ref>
In 2004, Sloane celebrated the addition of the 100,000th sequence to the database, {{OEIS link|A100000}}, which counts the marks on the [[Ishango bone]]. In 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki was created to simplify the collaboration of the OEIS editors and contributors.<ref>{{cite web | url = http://oeisf.org/announcementNov2010.txt | title = New version of OEIS | date = 2010-11-17 | author = Neil Sloane | access-date = 2011-01-21 | archive-date = 2016-02-07 | archive-url = https://web.archive.org/web/20160207093721/http://oeisf.org/announcementNov2010.txt | url-status = dead }}</ref> The 200,000th sequence, {{OEIS link|A200000}}, was added to the database in November 2011; it was initially entered as A200715, and moved to A200000 after a week of discussion on the SeqFan mailing list,<ref>{{cite web|url=http://list.seqfan.eu/pipermail/seqfan/2011-November/015853.html|title=<nowiki>[seqfan]</nowiki> A200000|author=Neil J. A. Sloane|work=SeqFan mailing list|access-date=2011-11-22|date=2011-11-14}}</ref><ref>{{cite web|url=http://list.seqfan.eu/pipermail/seqfan/2011-November/015926.html|author=Neil J. A. Sloane|work=SeqFan mailing list|title=<nowiki>[seqfan]</nowiki> A200000 chosen|date=2011-11-22|access-date=2011-11-22}}</ref> following a proposal by OEIS Editor-in-Chief [[Charles Greathouse]] to choose a special sequence for A200000.<ref>{{cite web|url=http://oeis.org/wiki/Suggested_Projects|work=OEIS wiki|title=Suggested Projects|access-date=2011-11-22}}</ref> A300000 was defined in February 2018, and by end of January 2023 the database contained more than 360,000 sequences.<ref name="MATH VALUES 2023 Sloane">{{cite web | title=Fifty Years of Integer Sequences | website=MATH VALUES | date=2023-12-01 | url=https://www.mathvalues.org/masterblog/fifty-years-of-integer-sequences | access-date=2023-12-04}}</ref><ref name="Sloane 2023 pp. 193–205">{{cite journal | last=Sloane | first=N. J. A. | title="A Handbook of Integer Sequences" Fifty Years Later | journal=The Mathematical Intelligencer | volume=45 | issue=3 | date=2023 | issn=0343-6993 | doi=10.1007/s00283-023-10266-6 | pages=193–205| doi-access=free | arxiv=2301.03149 }}</ref>

== Non-integers ==
Besides integer sequences, the OEIS also catalogs sequences of [[fraction]]s, the digits of [[transcendental number]]s, [[complex number]]s and so on by transforming them into integer sequences.
Sequences of fractions are represented by two sequences (named with the keyword 'frac'): the sequence of numerators and the sequence of denominators. For example, the fifth-order [[Farey sequence]], <math>\textstyle {1 \over 5}, {1 \over 4}, {1 \over 3}, {2 \over 5}, {1 \over 2}, {3 \over 5}, {2 \over 3}, {3 \over 4}, {4 \over 5}</math>, is catalogued as the numerator sequence 1, 1, 1, 2, 1, 3, 2, 3, 4 ({{OEIS link|A006842}}) and the denominator sequence 5, 4, 3, 5, 2, 5, 3, 4, 5 ({{OEIS link|A006843}}).
Important [[irrational number]]s such as π = 3.1415926535897... are catalogued under representative integer sequences such as [[decimal]] expansions (here 3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 3, 8, 4, 6, 2, 6, 4, 3, 3, 8, 3, 2, 7, 9, 5, 0, 2, 8, 8, ... ({{OEIS link|A000796}})), [[binary number|binary]] expansions (here 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, ... ({{OEIS link|A004601}})), or [[simple continued fraction|continued fraction expansions]] (here 3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, ... ({{OEIS link|A001203}})).

== Conventions ==
The OEIS was limited to plain [[ASCII]] text until 2011, and it still uses a linear form of conventional mathematical notation (such as ''f''(''n'') for [[function (mathematics)|functions]], ''n'' for running [[variable (mathematics)|variables]], etc.). [[Greek alphabet|Greek letters]] are usually represented by their full names, ''e.g.'', mu for μ, phi for φ.
Every sequence is identified by the letter A followed by six digits, almost always referred to with leading zeros, ''e.g.'', A000315 rather than A315.
Individual terms of sequences are separated by commas. Digit groups are not separated by commas, periods, or spaces.
In comments, formulas, etc., <code>a(n)</code> represents the ''n''th term of the sequence.

=== Special meaning of zero ===
<!-- This part needs a discussion on [[Imputation (statistics)|imputation]] of data. -->
[[Zero]] is often used to represent non-existent sequence elements. For example, {{OEIS link|A104157}} enumerates the "smallest [[prime number|prime]] of ''n''<sup>2</sup> consecutive primes to form an ''n'' × ''n'' [[magic square]] of least [[magic constant]], or 0 if no such magic square exists." The value of ''a''(1) (a 1 × 1 magic square) is 2; ''a''(3) is 1480028129. But there is no such 2 × 2 magic square, so ''a''(2) is 0. This special usage has a solid mathematical basis in certain counting functions; for example, the [[totient]] valence function ''N''<sub>φ</sub>(''m'') ({{OEIS link|A014197}}) counts the solutions of φ(''x'') = ''m''. There are 4 solutions for 4, but no solutions for 14, hence ''a''(14) of A014197 is 0—there are no solutions.

Other values are also used, most commonly −1 (see {{OEIS link|A000230}} or {{OEIS link|A094076}}).

=== Lexicographical ordering ===
The OEIS maintains the [[lexicographical order]] of the sequences, so each sequence has a predecessor and a successor (its "context").<ref>{{cite web|title=Welcome: Arrangement of the Sequences in Database|url=https://oeis.org/wiki/Welcome#Arrangement_of_the_Sequences_in_Database|website=OEIS Wiki|access-date=2016-05-05}}</ref> OEIS normalizes the sequences for lexicographical ordering, (usually) ignoring all initial zeros and ones, and also the [[sign (mathematics)|sign]] of each element. Sequences of [[weight distribution]] codes often omit periodically recurring zeros.<!-- Unclear, please rephrase. -->

For example, consider: the [[prime number]]s, the [[palindromic prime]]s, the [[Fibonacci number|Fibonacci sequence]], the [[lazy caterer's sequence]], and the coefficients in the [[series expansion]] of <math>\textstyle {{\zeta(n + 2)} \over {\zeta(n)}}</math>. In OEIS lexicographic order, they are:
* Sequence #1: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, ... {{OEIS link|id=A000040}}
* Sequence #2: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, ... {{OEIS link|id=A002385}}
* Sequence #3: {{bgcolor|lightpink|0, 1, 1,}} 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, ... {{OEIS link|id=A000045}}
* Sequence #4: {{bgcolor|lightpink|1,}} 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, ... {{OEIS link|id=A000124}}
* Sequence #5: {{bgcolor|lightpink|1,}} {{bgcolor|lightpink|−}}3, {{bgcolor|lightpink|−}}8, {{bgcolor|lightpink|−}}3, {{bgcolor|lightpink|−}}24, 24, {{bgcolor|lightpink|−}}48, {{bgcolor|lightpink|−}}3, {{bgcolor|lightpink|−}}8, 72, {{bgcolor|lightpink|−}}120, 24, {{bgcolor|lightpink|−}}168, 144, ... {{OEIS link|id=A046970}}
whereas unnormalized lexicographic ordering would order these sequences thus: #3, #5, #4, #1, #2.

==Self-referential sequences==
<!-- This section is linked from A053873 -->
Very early in the history of the OEIS, sequences defined in terms of the numbering of sequences in the OEIS itself were proposed. "I resisted adding these sequences for a long time, partly out of a desire to maintain the dignity of the database, and partly because A22 was only known to 11 terms!", Sloane reminisced.<ref>{{cite web | first = N. J. A. | last = Sloane | url = http://neilsloane.com/doc/sg.pdf | archive-url = https://web.archive.org/web/20180517140606/http://neilsloane.com/doc/sg.pdf | url-status = dead | archive-date = 2018-05-17 | title = My favorite integer sequences | page = 10 }}</ref>
One of the earliest self-referential sequences Sloane accepted into the OEIS was {{OEIS link|A031135}} (later {{OEIS link|A091967}}) "''a''(''n'') = ''n''-th term of sequence A<sub>''n''</sub> or −1 if A<sub>''n''</sub> has fewer than ''n'' terms". This sequence spurred progress on finding more terms of {{OEIS link|A000022}}.
{{OEIS link|A100544}} lists the first term given in sequence A<sub>''n''</sub>, but it needs to be updated from time to time because of changing opinions on offsets. Listing instead term ''a''(1) of sequence A<sub>''n''</sub> might seem a good alternative if it were not for the fact that some sequences have offsets of 2 and greater.
This line of thought leads to the question "Does sequence A<sub>''n''</sub> contain the number ''n''?" and the sequences {{OEIS link|A053873}}, "Numbers ''n'' such that OEIS sequence A<sub>''n''</sub> contains ''n''", and {{OEIS link|A053169}}, "''n'' is in this sequence [[if and only if]] ''n'' is not in sequence A<sub>''n''</sub>". Thus, the [[composite number]] 2808 is in A053873 because {{OEIS link|A002808}} is the sequence of composite numbers, while the non-prime 40 is in A053169 because it is not in {{OEIS link|id=A000040}}, the prime numbers. Each ''n'' is a member of exactly one of these two sequences, and in principle it can be determined ''which'' sequence each ''n'' belongs to, with two exceptions (related to the two sequences themselves):
*It cannot be determined whether 53873 is a member of A053873 or not. If it is in the sequence then by definition it should be; if it is not in the sequence then (again, by definition) it should not be. Nevertheless, either decision would be consistent, and would also resolve the question of whether 53873 is in A053169.
*It can be proved that 53169 [[principle of contradiction|both is and is not]] a member of A053169. If it is in the sequence then by definition it should not be; if it is not in the sequence then (again, by definition) it should be. This is a form of [[Russell's paradox]]. Hence it is also not possible to answer if 53169 is in A053873.

==Abridged example of a comprehensive entry==
This entry, {{OEIS link|A046970}}, was chosen because it comprehensively contains every OEIS field, filled.<ref>{{cite web |url=https://oeis.org/eishelp2.html |title=Explanation of Terms Used in Reply From |publisher=OEIS |author=N.J.A. Sloane |author-link=Neil Sloane}}</ref>

<syntaxhighlight lang="mathematica" style="overflow: auto;">
A046970     Dirichlet inverse of the Jordan function J_2 (A007434).
            1, -3, -8, -3, -24, 24, -48, -3, -8, 72, -120, 24, -168, 144, 192, -3, -288, 24, -360, 72, 384, 360, -528, 24, -24, 504, -8, 144, -840, -576, -960, -3, 960, 864, 1152, 24, -1368, 1080, 1344, 72, -1680, -1152, -1848, 360, 192, 1584, -2208, 24, -48, 72, 2304, 504, -2808, 24, 2880, 144, 2880, 2520, -3480, -576 
OFFSET	    1,2

COMMENTS	B(n+2) = -B(n)*((n+2)*(n+1)/(4*Pi^2))*z(n+2)/z(n) = -B(n)*((n+2)*(n+1)/(4*Pi^2)) * Sum_{j>=1} a(j)/j^(n+2).
            Apart from signs also Sum_{d|n} core(d)^2*mu(n/d) where core(x) is the squarefree part of x. - Benoit Cloitre, May 31 2002
REFERENCES	M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, 1965, pp. 805-811.
            T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986, p. 48.
LINKS	    Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
            M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
            P. G. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 (516) (2005) 403-408.
            Paul W. Oxby, A Function Based on Chebyshev Polynomials as an Alternative to the Sinc Function in FIR Filter Design, arXiv:2011.10546 [eess.SP], 2020.
            Wikipedia, Riemann zeta function.
FORMULA	    Multiplicative with a(p^e) = 1 - p^2.
            a(n) = Sum_{d|n} mu(d)*d^2.
            abs(a(n)) = Product_{p prime divides n} (p^2 - 1). - Jon Perry, Aug 24 2010
            From Wolfdieter Lang, Jun 16 2011: (Start)
            Dirichlet g.f.: zeta(s)/zeta(s-2).
            a(n) = J_{-2}(n)*n^2, with the Jordan function J_k(n), with J_k(1):=1. See the Apostol reference, p. 48. exercise 17. (End)
            a(prime(n)) = -A084920(n). - R. J. Mathar, Aug 28 2011
            G.f.: Sum_{k>=1} mu(k)*k^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 15 2017
EXAMPLE	    a(3) = -8 because the divisors of 3 are {1, 3} and mu(1)*1^2 + mu(3)*3^2 = -8.
            a(4) = -3 because the divisors of 4 are {1, 2, 4} and mu(1)*1^2 + mu(2)*2^2 + mu(4)*4^2 = -3.
            E.g., a(15) = (3^2 - 1) * (5^2 - 1) = 8*24 = 192. - Jon Perry, Aug 24 2010
            G.f. = x - 3*x^2 - 8*x^3 - 3*x^4 - 24*x^5 + 24*x^6 - 48*x^7 - 3*x^8 - 8*x^9 + ...
MAPLE	    Jinvk := proc(n, k) local a, f, p ; a := 1 ; for f in ifactors(n)[2] do p := op(1, f) ; a := a*(1-p^k) ; end do: a ; end proc:
            A046970 := proc(n) Jinvk(n, 2) ; end proc: # R. J. Mathar, Jul 04 2011
MATHEMATICA	muDD[d_] := MoebiusMu[d]*d^2; Table[Plus @@ muDD[Divisors[n]], {n, 60}] (Lopez)
            Flatten[Table[{ x = FactorInteger[n]; p = 1; For[i = 1, i <= Length[x], i++, p = p*(1 - x[[i]][[1]]^2)]; p}, {n, 1, 50, 1}]] (* Jon Perry, Aug 24 2010 *)
            a[ n_] := If[ n < 1, 0, Sum[ d^2 MoebiusMu[ d], {d, Divisors @ n}]] (* Michael Somos, Jan 11 2014 *)
            a[ n_] := If[ n < 2, Boole[ n == 1], Times @@ (1 - #[[1]]^2 & /@ FactorInteger @ n)] (* Michael Somos, Jan 11 2014 *)
PROG	    (PARI) A046970(n)=sumdiv(n, d, d^2*moebius(d)) \\ Benoit Cloitre
            (Haskell)
            a046970 = product . map ((1 -) . (^ 2)) . a027748_row
            -- Reinhard Zumkeller, Jan 19 2012
            (PARI) {a(n) = if( n<1, 0, direuler( p=2, n, (1 - X*p^2) / (1 - X))[n])} /* Michael Somos, Jan 11 2014 */
CROSSREFS	Cf. A007434, A027641, A027642, A063453, A023900.
            Cf. A027748.
            Sequence in context: A144457 A220138 A146975 * A322360 A058936 A280369
            Adjacent sequences:  A046967 A046968 A046969 * A046971 A046972 A046973
KEYWORD	    sign,easy,mult
AUTHOR	    Douglas Stoll, dougstoll(AT)email.msn.com
EXTENSIONS	Corrected and extended by Vladeta Jovovic, Jul 25 2001
            Additional comments from Wilfredo Lopez (chakotay147138274(AT)yahoo.com), Jul 01 2005
</syntaxhighlight>

===Entry fields===
; ID number
: Every sequence in the OEIS has a [[serial number]], a six-digit positive [[integer]], prefixed by A (and zero-padded on the left prior to November 2004). The letter "A" stands for "absolute". Numbers are either assigned by the editor(s) or by an A number dispenser, which is handy for when contributors wish to send in multiple related sequences at once and be able to create cross-references. An A number from the dispenser expires a month from issue if not used. But as the following table of arbitrarily selected sequences shows, the rough correspondence holds.

{| class="wikitable" align="center"
|-
!{{OEIS link|A059097}}
|Numbers ''n'' such that the [[binomial coefficient]] ''C''(2''n'',&nbsp;''n'') is not [[divisible]] by the [[square (algebra)|square]] of an [[parity (mathematics)|odd]] prime.
|{{nowrap|Jan 1, 2001}}
|-
!{{OEIS link|A060001}}
|[[Fibonacci number|Fibonacci]](''n'')!.
|{{nowrap|Mar 14, 2001}}
|-
!{{OEIS link|A066288}}
|Number of 3-dimensional [[polyomino]]es (or [[polycube]]s) with ''n'' cells and symmetry group of [[order (group theory)|order]] exactly 24.
|{{nowrap|Jan 1, 2002}}
|-
!{{OEIS link|A075000}}
|Smallest number such that ''n'' · ''a''(''n'') is a concatenation of ''n'' consecutive integers ...
|{{nowrap|Aug 31, 2002}}
|-
!{{OEIS link|A078470}}
|Continued fraction for ''ζ''(3/2)
|{{nowrap|Jan 1, 2003}}
|-
!{{OEIS link|A080000}}
|Number of permutations satisfying −''k''&nbsp;≤&nbsp;''p''(''i'')&nbsp;−&nbsp;''i''&nbsp;≤&nbsp;''r'' and ''p''(''i'')&nbsp;−&nbsp;''i''
|{{nowrap|Feb 10, 2003}}
|-
!{{OEIS link|A090000}}
|Length of longest contiguous block of 1s in binary expansion of ''n''th prime.
|{{nowrap|Nov 20, 2003}}
|-
!{{OEIS link|A091345}}
|Exponential convolution of A069321(''n'') with itself, where we set A069321(0)&nbsp;=&nbsp;0.
|{{nowrap|Jan 1, 2004}}
|-
!{{OEIS link|A100000}}
|Marks from the 22000-year-old [[Ishango bone]] from the Congo.
|{{nowrap|Nov 7, 2004}}
|-
!{{OEIS link|A102231}}
|Column 1 of triangle A102230, and equals the convolution of A032349 with A032349 shift right.
|{{nowrap|Jan 1, 2005}}
|-
!{{OEIS link|A110030}}
|Number of consecutive integers starting with ''n'' needed to sum to a Niven number.
|{{nowrap|Jul 8, 2005}}
|-
!{{OEIS link|A112886}}
|Triangle-free positive integers.
|{{nowrap|Jan 12, 2006}}
|-
!{{OEIS link|A120007}}
|[[Möbius transform]] of sum of prime [[divisor|factors]] of ''n'' with multiplicity.
|{{nowrap|Jun 2, 2006}}
|}
: Even for sequences in the book predecessors to the OEIS, the ID numbers are not the same. The 1973 ''Handbook of Integer Sequences'' contained about 2400 sequences, which were numbered by lexicographic order (the letter N plus four digits, zero-padded where necessary), and the 1995 ''Encyclopedia of Integer Sequences'' contained 5487 sequences, also numbered by lexicographic order (the letter M plus 4 digits, zero-padded where necessary). These old M and N numbers, as applicable, are contained in the ID number field in parentheses after the modern A number.
; Sequence data
: The sequence field lists the numbers themselves, to about 260 characters.<ref>{{Cite web|url=http://oeis.org/wiki/Style_Sheet|title=OEIS Style sheet}}</ref> More terms of the sequences can be provided in so-called B-files.<ref>{{Cite web|url=http://oeis.org/wiki/B-files|title=B-Files}}</ref> The sequence field makes no distinction between sequences that are finite but still too long to display and sequences that are infinite; instead, the keywords "fini", "full", and "more" are used to distinguish such sequences. To determine to which ''n'' the values given correspond, see the offset field, which gives the ''n'' for the first term given.
; Name
: The name field usually contains the most common name for the sequence, and sometimes also the formula. For example, 1, 8, 27, 64, 125, 216, 343, 512, ({{OEIS link|A000578}}) is named "The [[cube (algebra)|cubes]]: a(n) = n^3.".
; Comments
: The comments field is for information about the sequence that does not quite fit in any of the other fields. The comments field often points out interesting relationships between different sequences and less obvious applications for a sequence. For example, Lekraj Beedassy in a comment to A000578 notes that the cube numbers also count the "total number of [[triangle]]s resulting from criss-crossing [[cevian]]s within a triangle so that two of its sides are each ''n''-partitioned", and Neil Sloane points out an unexpected relationship between [[centered hexagonal number]]s ({{OEIS link|A003215}}) and second [[Bessel polynomials]] ({{OEIS link|A001498}}) in a comment to A003215.
; References
: References to printed documents (books, papers, ...).
; Links
: Links, i.e. [[Uniform Resource Locator|URLs]], to online resources. These may be:
:# references to applicable articles in journals
:# links to the index
:# links to text files which hold the sequence terms (in a two column format) over a wider range of indices than held by the main database lines
:# links to images in the local database directories which often provide combinatorial background related to [[graph theory]]
:# others related to computer codes, more extensive tabulations in specific research areas provided by individuals or research groups
; Formula
: Formulas, [[recurrence relation|recurrences]], [[generating function]]s, etc. for the sequence.
; Example
: Some examples of sequence member values.
; Maple
: [[Maple computer algebra system|Maple]] code.
; Mathematica
: [[Wolfram Language]] code.
; Program
: Originally [[Maple computer algebra system|Maple]] and [[Mathematica]] were the preferred programs for calculating sequences in the OEIS, each with their own field labels. {{As of|2016}}, Mathematica was the most popular choice with 100,000 Mathematica programs followed by 50,000 [[PARI/GP]] programs, 35,000 Maple programs, and 45,000 in other languages.
: As for any other part of the record, if there is no name given, the contribution (here: program) was written by the original submitter of the sequence.
; Crossrefs
: Sequence cross-references originated by the original submitter are usually denoted by "[[Cf.]]"
: Except for new sequences, the "see also" field also includes information on the lexicographic order of the sequence (its "context") and provides links to sequences with close A numbers (A046967, A046968, A046969, A046971, A046972, A046973, in our example). The following table shows the context of our example sequence, A046970:

{| class="wikitable" align="center"
|-
!{{OEIS link|A016623}}
|3, 8, 3, 9, 4, 5, 2, 3, 1, 2, ...
|Decimal expansion of [[natural logarithm|ln]](93/2).
|-
!{{OEIS link|A046543}}
|1, 1, 1, 3, 8, 3, 10, 1, 110, 3, 406, 3
|First numerator and then denominator of the central<br />elements of the 1/3-Pascal triangle (by row).
|-
!{{OEIS link|A035292}}
|1, 3, 8, 3, 12, 24, 16, 3, 41, 36, 24, ...
|Number of similar sublattices of '''Z'''<sup>4</sup> of index ''n''<sup>2</sup>.
|-
!{{OEIS link|A046970}}
|1, −3, −8, −3, −24, 24, −48, −3, −8, 72, ...
|Generated from [[Riemann zeta function]]...
|-
!{{OEIS link|A058936}}
|0, 1, 3, 8, 3, 30, 20, 144, 90, 40, 840,<br />504, 420, 5760, 3360, 2688, 1260
|Decomposition of Stirling's ''S''(''n'', 2) based on<br />associated numeric partitions.
|-
!{{OEIS link|A002017}}
|1, 1, 1, 0, −3, −8, −3, 56, 217, 64, −2951, −12672, ...
|Expansion of&nbsp;[[exponential function|exp]]([[sine|sin]] ''x'').
|-
!{{OEIS link|A086179}}
|3, 8, 4, 1, 4, 9, 9, 0, 0, 7, 5, 4, 3, 5, 0, 7, 8
|Decimal expansion of upper bound for the r-values<br />supporting stable period-3 orbits in the [[logistic map]].
|}
; Keyword
: The OEIS has its own [[lexicon]]: a standard set of mostly four-letter keywords which [[taxonomy|characterizes]] each sequence:<ref name="terms-explanation">{{cite encyclopedia | url = http://oeis.org/classic/eishelp2.html | encyclopedia = On-Line Encyclopedia of Integer Sequences | title = Explanation of Terms Used in Reply From}}</ref>
:*'''allocated''' – An A-number which has been set aside for a user but for which the entry has not yet been approved (and perhaps not yet written).
:*'''base''' – The results of the calculation depend on a specific [[Positional notation|positional base]]. For example, 2, 3, 5, 7, 11, 101, 131, 151, 181 ... {{OEIS link|A002385}} are prime numbers regardless of base, but they are [[palindromic prime|palindromic]] specifically in base 10. Most of them are not palindromic in binary. Some sequences rate this keyword depending on how they are defined. For example, the [[Mersenne prime]]s 3, 7, 31, 127, 8191, 131071, ... {{OEIS link|A000668}} does not rate "base" if defined as "primes of the form 2^n − 1". However, defined as "[[repunit]] primes in binary," the sequence would rate the keyword "base".
:* '''bref''' – "sequence is too short to do any analysis with", for example, {{OEIS link|A079243}}, the number of [[isomorphism class]]es of [[associative]] non-[[commutative]] non-anti-associative [[anti-commutative]] closed [[binary operation]]s on a [[set (mathematics)|set]] of order ''n''.
:* '''changed''' The sequence is changed in the last two weeks.
:* '''cofr''' – The sequence represents a [[continued fraction]], for example the continued fraction expansion of ''e'' ({{OEIS link|A003417}}) or π ({{OEIS link|A001203}}).
:* '''cons''' – The sequence is a decimal expansion of a [[mathematical constant]], such as ''e'' ({{OEIS link|A001113}}) or π ({{OEIS link|A000796}}).
:* '''core''' – A sequence that is of foundational importance to a branch of mathematics, such as the prime numbers ({{OEIS link|A000040}}), the Fibonacci sequence ({{OEIS link|A000045}}), etc.
:* '''dead''' – This keyword used for erroneous sequences that have appeared in papers or books, or for duplicates of existing sequences. For example, {{OEIS link|A088552}} is the same as {{OEIS link|A000668}}.
:* '''dumb''' – One of the more subjective keywords, for "unimportant sequences," which may or may not directly relate to mathematics, such as [[popular culture]] references, arbitrary sequences from Internet puzzles, and sequences related to [[numeric keypad]] entries. {{OEIS link|A001355}}, "Mix digits of pi and e" is one example of lack of importance, and {{OEIS link|A085808}}, "Price is Right wheel" (the sequence of numbers on the [[Showcase Showdown]] wheel used in the U.S. game show ''[[The Price Is Right (U.S. game show)|The Price Is Right]]'') is an example of a non-mathematics-related sequence, kept mainly for trivia purposes.<ref>The person who submitted A085808 did so as an example of a sequence that should not have been included in the OEIS. Sloane added it anyway, surmising that the sequence "might appear one day on a quiz."</ref>
:* '''easy''' – The terms of the sequence can be easily calculated. Perhaps the sequence most deserving of this keyword is 1, 2, 3, 4, 5, 6, 7, ... {{OEIS link|A000027}}, where each term is 1 more than the previous term. The keyword "easy" is sometimes given to sequences "primes of the form ''f''(''m'')" where ''f''(''m'') is an easily calculated function. (Though even if ''f''(''m'') is easy to calculate for large ''m'', it might be very difficult to determine if ''f''(''m'') is prime).
:* '''eigen''' – A sequence of [[eigenvalue]]s.
:* '''fini''' – The sequence is finite, although it might still contain more terms than can be displayed. For example, the sequence field of {{OEIS link|A105417}} shows only about a quarter of all the terms, but a comment notes that the last term is 3888.
:* '''frac''' – A sequence of either numerators or denominators of a sequence of fractions representing [[rational number]]s. Any sequence with this keyword ought to be cross-referenced to its matching sequence of numerators or denominators, though this may be dispensed with for sequences of [[Egyptian fraction]]s, such as {{OEIS link|A069257}}, where the sequence of numerators would be {{OEIS link|A000012}}. This keyword should not be used for sequences of continued fractions; cofr should be used instead for that purpose.
:* '''full''' – The sequence field displays the complete sequence. If a sequence has the keyword "full", it should also have the keyword "fini". One example of a finite sequence given in full is that of the [[Supersingular prime (moonshine theory)|supersingular prime]]s {{OEIS link|A002267}}, of which there are precisely fifteen.
:* '''hard''' – The terms of the sequence cannot be easily calculated, even with raw number crunching power. This keyword is most often used for sequences corresponding to unsolved problems, such as "How many [[n-sphere|''n''-spheres]] can touch another ''n''-sphere of the same size?" {{OEIS link|A001116}} lists the first ten known solutions.
:* '''hear''' – A sequence with a graph audio deemed to be "particularly interesting and/or beautiful", some examples are collected at the [https://oeis.org/play.html OEIS site].
:* '''less''' – A "less interesting sequence".
:* '''look''' – A sequence with a graph visual deemed to be "particularly interesting and/or beautiful". Two examples out of several thousands are [https://oeis.org/A331124/graph A331124] [https://oeis.org/A347347/graph A347347].
:* '''more''' – More terms of the sequence are wanted. Readers can submit an extension.
:* '''mult''' – The sequence corresponds to a [[multiplicative function]]. Term ''a''(1) should be 1, and term ''a''(''mn'') can be calculated by multiplying ''a''(''m'') by ''a''(''n'') if ''m'' and ''n'' are [[coprime]]. For example, in {{OEIS link|A046970}}, ''a''(12) = ''a''(3)''a''(4) = −8 × −3.
:* '''new''' – For sequences that were added in the last couple of weeks, or had a major extension recently. This keyword is not given a checkbox in the Web form for submitting new sequences; Sloane's program adds it by default where applicable.
:* '''nice''' – Perhaps the most subjective keyword of all, for "[[Mathematical beauty|exceptionally nice sequences]]."
:* '''nonn''' – The sequence consists of nonnegative integers (it may include zeroes). No distinction is made between sequences that consist of nonnegative numbers only because of the chosen offset (e.g., ''n''<sup>3</sup>, the cubes, which are all nonnegative from ''n'' = 0 forwards) and those that by definition are completely nonnegative (e.g., ''n''<sup>2</sup>, the squares).
:* '''obsc''' – The sequence is considered obscure and needs a better definition.
:* '''recycled''' – When the editors agree that a new proposed sequence is not worth adding to the OEIS, an editor blanks the entry leaving only the keyword line with keyword:recycled. The A-number then becomes available for allocation for another new sequence.
:* '''sign''' – Some (or all) of the values of the sequence are negative. The entry includes both a Signed field with the signs and a Sequence field consisting of all the values passed through the [[absolute value]] function.
:* '''tabf''' – "An irregular (or funny-shaped) array of numbers made into a sequence by reading it row by row." For example, {{OEIS link|A071031}}, "Triangle read by rows giving successive states of [[cellular automaton]] generated by "rule 62."
:* '''tabl''' – A sequence obtained by reading a geometric arrangement of numbers, such as a triangle or square, row by row. The quintessential example is [[Pascal's triangle]] read by rows, {{OEIS link|A007318}}.
:* '''uned''' – The sequence has not been edited but it could be worth including in the OEIS. The sequence may contain computational or typographical errors. Contributors are encouraged to edit these sequences.
:* '''unkn''' – "Little is known" about the sequence, not even the formula that produces it. For example, {{OEIS link|A072036}}, which was presented to the [[Internet Oracle]] to ponder.
:* '''walk''' – "Counts walks (or [[self-avoiding walk|self-avoiding paths]])."
:* '''word''' – Depends on the words of a specific language. For example, zero, one, two, three, four, five, etc. For example, 4, 3, 3, 5, 4, 4, 3, 5, 5, 4, 3, 6, 6, 8, 8, 7, 7, 9, 8, 8 ... {{OEIS link|A005589}}, "Number of letters in the English name of ''n'', excluding spaces and hyphens."
: Some keywords are mutually exclusive, namely: core and dumb, easy and hard, full and more, less and nice, and nonn and sign.
; Offset
: The offset is the index of the first term given. For some sequences, the offset is obvious. For example, if we list the sequence of square numbers as 0, 1, 4, 9, 16, 25 ..., the offset is 0; while if we list it as 1, 4, 9, 16, 25 ..., the offset is 1. The default offset is 0, and most sequences in the OEIS have offset of either 0 or 1. Sequence {{OEIS link|A073502}}, the [[magic constant]] for ''n'' × ''n'' [[magic square]] with prime entries (regarding 1 as a prime) with smallest row sums, is an example of a sequence with offset 3, and {{OEIS link|A072171}}, "Number of stars of visual magnitude ''n''." is an example of a sequence with offset −1. Sometimes there can be disagreement over what the initial terms of the sequence are, and correspondingly what the offset should be. In the case of the [[lazy caterer's sequence]], the maximum number of pieces you can cut a pancake into with ''n'' cuts, the OEIS gives the sequence as 1, 2, 4, 7, 11, 16, 22, 29, 37, ... {{OEIS link|A000124}}, with offset 0, while [[Mathworld]] gives the sequence as 2, 4, 7, 11, 16, 22, 29, 37, ... (implied offset 1). It can be argued that making no cuts to the pancake is technically a number of cuts, namely ''n'' = 0, but it can also be argued that an uncut pancake is irrelevant to the problem. Although the offset is a required field, some contributors do not bother to check if the default offset of 0 is appropriate to the sequence they are sending in. The internal format actually shows two numbers for the offset. The first is the number described above, while the second represents the index of the first entry (counting from 1) that has an absolute value greater than 1. This second value is used to speed up the process of searching for a sequence. Thus {{OEIS link|A000001}}, which starts 1, 1, 1, 2 with the first entry representing ''a''(1) has '''1, 4''' as the internal value of the offset field.
; Author(s)
: The author(s) of the sequence is (are) the person(s) who submitted the sequence, even if the sequence has been known since ancient times. The name of the submitter(s) is given first name (spelled out in full), middle initial(s) (if applicable) and last name; this in contrast to the way names are written in the reference fields. The e-mail address of the submitter is also given before 2011, with the @ character replaced by "(AT)" with some exceptions such as for associate editors or if an e-mail address does not exist. Now it has been the policy for OEIS not to display e-mail addresses in sequences. For most sequences after A055000, the author field also includes the date the submitter sent in the sequence.
; Extension
: Names of people who extended (added more terms to) the sequence or corrected terms of a sequence, followed by date of extension.

==Sloane's gap==
[[File:Sloanes gap.png|thumb|Plot of Sloane's Gap: number of occurrences (''y'' log scale) of each integer (''x'' scale) in the OEIS database]]In 2009, the OEIS database was used by Philippe Guglielmetti to measure the "importance" of each integer number.<ref>{{cite web|last1=Guglielmetti|first1=Philippe|title=Chasse aux nombres acratopèges|url=http://www.drgoulu.com/2008/08/24/nombres-acratopeges|website=Pourquoi Comment Combien|date=24 August 2008 |language=fr}}</ref> The result shown in the plot on the right shows a clear "gap" between two distinct point clouds,<ref>{{cite web|last1=Guglielmetti|first1=Philippe|title=La minéralisation des nombres|url=http://www.drgoulu.com/2009/04/18/nombres-mineralises|website=Pourquoi Comment Combien|date=18 April 2009 |access-date=25 December 2016|language=fr}}</ref> the "[[Interesting number paradox|uninteresting numbers]]" (blue dots) and the "interesting" numbers that occur comparatively more often in sequences from the OEIS. It contains essentially prime numbers (red), numbers of the form ''a''<sup>''n''</sup> (green) and [[highly composite number]]s (yellow). This phenomenon was studied by [[Nicolas Gauvrit]], [[Jean-Paul Delahaye]] and Hector Zenil who explained the speed of the two clouds in terms of algorithmic complexity and the gap by social factors based on an artificial preference for sequences of primes, [[parity (mathematics)|even]] numbers, geometric and Fibonacci-type sequences and so on.<ref>{{cite journal|last1=Gauvrit|first1=Nicolas|last2=Delahaye|first2=Jean-Paul|last3=Zenil|first3=Hector|title=Sloane's Gap. Mathematical and Social Factors Explain the Distribution of Numbers in the OEIS|journal=Journal of Humanistic Mathematics|date=2011|volume=3|pages=3–19|doi=10.5642/jhummath.201301.03|url=https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1048&context=jhm|arxiv=1101.4470|bibcode=2011arXiv1101.4470G|s2cid=22115501}}</ref> Sloane's gap was featured on a [[Numberphile]] video in 2013.<ref>{{cite web |url= https://www.youtube.com/watch?v=_YysNM2JoFo | archive-url=https://ghostarchive.org/varchive/youtube/20211117/_YysNM2JoFo| archive-date=2021-11-17 | url-status=live|title= Sloane's Gap |work=[[Numberphile]] |format= video |quote= With Dr. James Grime, [[University of Nottingham]] |date=2013-10-15}}{{cbignore}}</ref>

==See also==
* [[List of OEIS sequences]]
* [[Abramowitz and Stegun]]

==Notes==
{{Reflist}}

==References==
*{{cite journal |first1=J.|last1=Borwein|author1-link=Jonathan Borwein|first2= R.|last2=Corless |url=http://www.cecm.sfu.ca/~jborwein/sloane/sloane.html |title=The Encyclopedia of Integer Sequences (N. J. A. Sloane and Simon Plouffe) |journal=SIAM Review |volume=38 |issue=2 |year=1996 |pages=333–337 |doi=10.1137/1038058}}
*{{cite journal |first=H.|last=Catchpole |url=http://abc.net.au/science/news/stories/s1209743.htm |title=Exploring the number jungle online |journal=ABC Science |publisher=[[Australian Broadcasting Corporation]] |year=2004}}
*{{cite journal |first=A.|last=Delarte |title=Mathematician reaches 100k milestone for online integer archive |journal=[[The South End]] |date=November 11, 2004 |page=5}}
*{{cite journal |first=B. |last=Hayes |author-link=Brian Hayes (scientist) |url=http://lacim.uqam.ca/~plouffe/articles/A%20Question%20of%20Numbers.pdf |title=A Question of Numbers |journal=[[American Scientist]] |volume=84 |issue=1 |pages=10–14 |year=1996 |bibcode=1996AmSci..84...10H |access-date=2010-06-01 |archive-date=2015-10-05 |archive-url=https://web.archive.org/web/20151005070524/http://lacim.uqam.ca/~plouffe/articles/A%20Question%20of%20Numbers.pdf |url-status=dead }}
*{{cite journal |first=I. |last=Peterson |author-link=Ivars Peterson |url=http://www.plouffe.fr/simon/OEIS/citations/Math%20Trek_%20Sequence%20Puzzles,%20Science%20News%20Online,%20May%2017,%202003.pdf |title=Sequence Puzzles |journal=[[Science News]] |volume=163 |year=2003 |issue=20 |access-date=2016-12-24 |archive-url=https://web.archive.org/web/20170510101432/http://www.plouffe.fr/simon/OEIS/citations/Math%20Trek_%20Sequence%20Puzzles,%20Science%20News%20Online,%20May%2017,%202003.pdf |archive-date=2017-05-10 |url-status=dead }}
*{{cite journal |first=J. |last=Rehmeyer |url=https://www.sciencenews.org/article/pattern-collector |title=The Pattern Collector&nbsp;— Science News |publisher=www.sciencenews.org |access-date=2010-08-08 |journal=[[Science News]] |year=2010 |archive-url=https://web.archive.org/web/20131014165834/https://www.sciencenews.org/article/pattern-collector |archive-date=2013-10-14 |url-status=dead }}

==Further reading==
* {{Citation | vauthors=((Roberts, S.)) | date=May 21, 2023 | website=The New York Times | title=What Number Comes Next? The Encyclopedia of Integer Sequences Knows. | url=https://www.nytimes.com/2023/05/21/science/math-puzzles-integer-sequences.html | access-date=21 May 2023}}
*{{Cite book
 | contribution-url = http://neilsloane.com/doc/sg.pdf
 | contribution = My favorite integer sequences
 | first = N. J. A.
 | last = Sloane
 | author-link = Neil Sloane
 | title = Sequences and their Applications (Proceedings of SETA '98)
 | editor1-first = C.
 | editor1-last = Ding
 | editor2-first = T.
 | editor2-last = Helleseth
 | editor3-first = H.
 | editor3-last = Niederreiter | editor3-link = Harald Niederreiter
 | publisher = Springer-Verlag
 | location = London
 | year = 1999
 | pages = 103–130
 | arxiv=math/0207175
| bibcode = 2002math......7175S
 }}
*{{Cite journal
 | url = https://www.ams.org/notices/200308/comm-sloane.pdf
 | first = N. J. A.
 | last = Sloane
 | author-link = Neil Sloane
 | title = The On-Line Encyclopedia of Integer Sequences
 | journal = Notices of the American Mathematical Society
 | volume = 50
 | issue = 8
 | year = 2003
 | pages = 912–915
 }}
*{{cite book |first1=N. J. A. |last1=Sloane|author1-link=Neil Sloane|first2= S.|last2=Plouffe|author2-link=Simon Plouffe |title=The Encyclopedia of Integer Sequences |url=http://oeis.org/book.html |publisher=Academic Press |location=San Diego |year=1995 |isbn=0-12-558630-2}}
*{{Cite journal
 | url = https://habr.com/en/articles/701208/
 | first1 = A. |last1 = Zabolotskii
 | title =  The On-Line Encyclopedia of Integer Sequences in 2021
 | journal = Mat. Pros. |series=Series 3
 | volume = 8
 | year = 2022
 | pages = 199–212
 }}
*{{Cite journal
 | url = https://www.ams.org/notices/201308/rnoti-p1034.pdf
 | doi = 10.1090/noti1029
 | arxiv = 1304.3866
 | first1 = Sara C. |last1=Billey|author1-link=Sara Billey|first2= Bridget E.|last2=Tenner|author2-link=Bridget Tenner
 | title =  Fingerprint databases for theorems
 | journal = Notices of the American Mathematical Society
 | volume = 60
 | issue = 8
 | year = 2013
 | pages = 1034–1039
 | bibcode = 2013arXiv1304.3866B
 | s2cid = 14435520
 }}

==External links==
{{commons category|OEIS}}
* {{official website|//oeis.org/}}
* [http://oeis.org/wiki/Main_Page Wiki] at OEIS

[[Category:On-Line Encyclopedia of Integer Sequences| ]]
[[Category:Mathematical databases]]
[[Category:Integer sequences|*]]
[[Category:Encyclopedias of mathematics]]
[[Category:Multilingual websites]]
[[Category:Mathematical projects]]
[[Category:20th-century encyclopedias]]
[[Category:21st-century encyclopedias]]
[[Category:American online encyclopedias]]