On-Line Encyclopedia of Integer Sequences

"OEIS" redirects here. For the birth defect known as OEIS complex, seeCloacal exstrophy.

On-Line Encyclopedia of Integer Sequences

Founded	1964(1964)
Predecessor(s)	Handbook of Integer Sequences,Encyclopedia of Integer Sequences
Created by	Neil Sloane
Chairman	Neil Sloane
President	Russ Cox
URL	oeis.org
Commercial	No
Registration	Optional
Launched	1996(1996)
Content license	Creative CommonsCC BY-SA4.0

TheOn-Line Encyclopedia of Integer Sequences(OEIS) is an online database ofinteger sequences.It was created and maintained byNeil Sloanewhile researching atAT&T Labs.He transferred theintellectual propertyand hosting of the OEIS to theOEIS Foundationin 2009.Sloane is the chairman of the OEIS Foundation.

OEIS records information on integer sequences of interest to both professional andamateurmathematicians,and is widely cited. As of February 2024,it contains over 370,000 sequences,and is growing by approximately 30 entries per day.

Each entry contains the leading terms of the sequence,keywords,mathematical motivations, literature links, and more, including the option to generate agraphor play amusicalrepresentation of the sequence. The database issearchableby keyword, bysubsequence,or by any of 16 fields.

History

Second edition of the book

Neil Sloanestarted collecting integer sequences as a graduate student in 1964 to support his work incombinatorics.The database was at first stored onpunched cards.He published selections from the database in book form twice:

1. A Handbook of Integer Sequences(1973,ISBN0-12-648550-X), containing 2,372 sequences inlexicographic orderand assigned numbers from 1 to 2372.
2. The Encyclopedia of Integer SequenceswithSimon Plouffe(1995,ISBN0-12-558630-2), 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) inA Handbook of Integer Sequencesas 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.

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 had reached 16,000 entries Sloane decided to go online—first as anemailservice (August 1994), and soon after as a website (1996). As a spin-off from the database work, Sloane founded theJournal of Integer Sequencesin 1998. 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 database. In 2004, Sloane celebrated the addition of the 100,000th sequence to the database,A100000,which counts the marks on theIshango bone.In 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 anOEIS wikiatOEIS.orgwas created to simplify the collaboration of the OEIS editors and contributors.The 200,000th sequence,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,following a proposal by OEIS Editor-in-ChiefCharles Greathouseto choose a special sequence for A200000.A300000 was defined in February 2018, and by end of January 2023 the database contained more than 360,000 sequences.

Non-integers

Besides integer sequences, the OEIS also catalogs sequences offractions,the digits oftranscendental numbers,complex numbersand 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-orderFarey sequence,$\textstyle {1 \over 5},{1 \over 4},{1 \over 3},{2 \over 5},{1 \over 2},{3 \over 5},{2 \over 3},{3 \over 4},{4 \over 5}$,is catalogued as the numerator sequence 1, 1, 1, 2, 1, 3, 2, 3, 4 (A006842) and the denominator sequence 5, 4, 3, 5, 2, 5, 3, 4, 5 (A006843). Importantirrational numberssuch as π = 3.1415926535897... are catalogued under representative integer sequences such asdecimalexpansions (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,... (A000796)),binaryexpansions (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,... (A004601)), orcontinued fractionexpansions (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,... (A001203)).

Conventions

The OEIS was limited to plainASCIItext until 2011, and it still uses a linear form of conventional mathematical notation (such asf(n) forfunctions,nfor runningvariables,etc.).Greek lettersare 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.,a(n)represents thenth term of the sequence.

Special meaning of zero

Zerois often used to represent non-existent sequence elements. For example,A104157enumerates the "smallestprimeofn²consecutive primes to form ann×nmagic squareof leastmagic constant,or 0 if no such magic square exists. "The value ofa(1) (a 1 × 1 magic square) is 2;a(3) is 1480028129. But there is no such 2 × 2 magic square, soa(2) is 0. This special usage has a solid mathematical basis in certain counting functions; for example, thetotientvalence functionN_φ(m) (A014197) counts the solutions of φ(x) =m.There are 4 solutions for 4, but no solutions for 14, hencea(14) of A014197 is 0—there are no solutions.

Other values are also used, most commonly −1 (seeA000230orA094076).

Lexicographical ordering

The OEIS maintains thelexicographical orderof the sequences, so each sequence has a predecessor and a successor (its "context" ).OEIS normalizes the sequences for lexicographical ordering, (usually) ignoring all initial zeros and ones, and also thesignof each element. Sequences ofweight distributioncodes often omit periodically recurring zeros.

For example, consider: theprime numbers,thepalindromic primes,theFibonacci sequence,thelazy caterer's sequence,and the coefficients in theseries expansionof$\textstyle {{\zeta (n+2)} \over {\zeta (n)}}$.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,...A000040
• Sequence #2: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929,...A002385
• Sequence #3:0, 1, 1,2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597,...A000045
• Sequence #4:1,2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154,...A000124
• Sequence #5:1,−3,−8,−3,−24, 24,−48,−3,−8, 72,−120, 24,−168, 144,...A046970

whereas unnormalized lexicographic ordering would order these sequences thus: #3, #5, #4, #1, #2.

Self-referential sequences

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. One of the earliest self-referential sequences Sloane accepted into the OEIS wasA031135(laterA091967) "a(n) =n-th term of sequence A_nor –1 if A_nhas fewer thannterms ". This sequence spurred progress on finding more terms ofA000022. A100544lists the first term given in sequence A_n,but it needs to be updated from time to time because of changing opinions on offsets. Listing instead terma(1) of sequence A_nmight 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_ncontain the numbern?"and the sequencesA053873,"Numbersnsuch that OEIS sequence A_ncontainsn",andA053169,"nis in this sequenceif and only ifnis not in sequence A_n".Thus, thecomposite number2808 is in A053873 becauseA002808is the sequence of composite numbers, while the non-prime 40 is in A053169 because it is not inA000040,the prime numbers. Eachnis a member of exactly one of these two sequences, and in principle it can be determinedwhichsequence eachnbelongs 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 53169both is and is nota 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 ofRussell's paradox.Hence it is also not possible to answer if 53169 is in A053873.

Abridged example of a typical entry

This entry,A046970,was chosen because it contains every field an OEIS entry can have.

A046970DirichletinverseoftheJordanfunctionJ_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
OFFSET1,2

COMMENTSB(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).
ApartfromsignsalsoSum_{d|n}core(d)^2*mu(n/d)wherecore(x)isthesquarefreepartofx.-BenoitCloitre,May312002
REFERENCESM.AbramowitzandI.A.Stegun,HandbookofMathematicalFunctions,DoverPublications,1965,pp.805-811.
T.M.Apostol,IntroductiontoAnalyticNumberTheory,Springer-Verlag,1986,p.48.
LINKSReinhardZumkeller,Tableofn,a(n)forn=1..10000
M.AbramowitzandI.A.Stegun,eds.,HandbookofMathematicalFunctions,NationalBureauofStandards,AppliedMath.Series55,TenthPrinting,1972[alternativescannedcopy].
P.G.Brown,Somecommentsoninversearithmeticfunctions,Math.Gaz.89(516)(2005)403-408.
PaulW.Oxby,AFunctionBasedonChebyshevPolynomialsasanAlternativetotheSincFunctioninFIRFilterDesign,arXiv:2011.10546[eess.SP],2020.
Wikipedia,Riemannzetafunction.
FORMULAMultiplicativewitha(p^e)=1-p^2.
a(n)=Sum_{d|n}mu(d)*d^2.
abs(a(n))=Product_{pprimedividesn}(p^2-1).-JonPerry,Aug242010
FromWolfdieterLang,Jun162011:(Start)
Dirichletg.f.:zeta(s)/zeta(s-2).
a(n)=J_{-2}(n)*n^2,withtheJordanfunctionJ_k(n),withJ_k(1):=1.SeetheApostolreference,p.48.exercise17.(End)
a(prime(n))=-A084920(n).-R.J.Mathar,Aug282011
G.f.:Sum_{k>=1}mu(k)*k^2*x^k/(1-x^k).-IlyaGutkovskiy,Jan152017
EXAMPLEa(3)=-8becausethedivisorsof3are{1,3}andmu(1)*1^2+mu(3)*3^2=-8.
a(4)=-3becausethedivisorsof4are{1,2,4}andmu(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.-JonPerry,Aug242010
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+...
MAPLEJinvk:=proc(n,k)locala,f,p;a:=1;forfinifactors(n)[2]dop:=op(1,f);a:=a*(1-p^k);enddo:a;endproc:
A046970:=proc(n)Jinvk(n,2);endproc:#R.J.Mathar,Jul042011
MATHEMATICAmuDD[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^2MoebiusMu[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))\\BenoitCloitre
(Haskell)
a046970=product.map((1-).(^2)).a027748_row
--ReinhardZumkeller,Jan192012
(PARI){a(n)=if(n<1,0,direuler(p=2,n,(1-X*p^2)/(1-X))[n])}/*MichaelSomos,Jan112014*/
CROSSREFSCf.A007434,A027641,A027642,A063453,A023900.
Cf.A027748.
Sequenceincontext:A144457A220138A146975*A322360A058936A280369
Adjacentsequences:A046967A046968A046969*A046971A046972A046973
KEYWORDsign,easy,mult
AUTHORDouglasStoll,dougstoll(AT)email.msn.com
EXTENSIONSCorrectedandextendedbyVladetaJovovic,Jul252001
AdditionalcommentsfromWilfredoLopez(chakotay147138274(AT)yahoo.com),Jul012005

Entry fields

ID number

Every sequence in the OEIS has aserial number,a six-digit positiveinteger,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.

A059097	Numbersnsuch that thebinomial coefficientC(2n,n) is notdivisibleby thesquareof anoddprime.	Jan 1, 2001
A060001	Fibonacci(n)!.	Mar 14, 2001
A066288	Number of 3-dimensionalpolyominoes(orpolycubes) withncells and symmetry group oforderexactly 24.	Jan 1, 2002
A075000	Smallest number such thatn·a(n) is a concatenation ofnconsecutive integers...	Aug 31, 2002
A078470	Continued fraction forζ(3/2)	Jan 1, 2003
A080000	Number of permutations satisfying −k≤p(i) −i≤randp(i) −i	Feb 10, 2003
A090000	Length of longest contiguous block of 1s in binary expansion ofnth prime.	Nov 20, 2003
A091345	Exponential convolution of A069321(n) with itself, where we set A069321(0) = 0.	Jan 1, 2004
A100000	Marks from the 22000-year-oldIshango bonefrom the Congo.	Nov 7, 2004
A102231	Column 1 of triangle A102230, and equals the convolution of A032349 with A032349 shift right.	Jan 1, 2005
A110030	Number of consecutive integers starting withnneeded to sum to a Niven number.	Jul 8, 2005
A112886	Triangle-free positive integers.	Jan 12, 2006
A120007	Möbius transformof sum of primefactorsofnwith multiplicity.	Jun 2, 2006

Even for sequences in the book predecessors to the OEIS, the ID numbers are not the same. The 1973Handbook of Integer Sequencescontained about 2400 sequences, which were numbered by lexicographic order (the letter N plus four digits, zero-padded where necessary), and the 1995Encyclopedia of Integer Sequencescontained 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.More terms of the sequences can be provided in so-called B-files.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 whichnthe values given correspond, see the offset field, which gives thenfor 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, (A000578) is named "Thecubes: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 oftrianglesresulting from criss-crossingcevianswithin a triangle so that two of its sides are eachn-partitioned, "while Neil Sloane points out the unexpected relationship betweencentered hexagonal numbers(A003215) and secondBessel polynomials(A001498) in a comment to A003215.

References

References to printed documents (books, papers,...).

Links

Links, i.e.URLs,to online resources. These may be:

1. references to applicable articles in journals
2. links to the index
3. 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
4. links to images in the local database directories which often provide combinatorial background related tograph theory
5. others related to computer codes, more extensive tabulations in specific research areas provided by individuals or research groups

Formula

Formulas,recurrences,generating functions,etc. for the sequence.

Example

Some examples of sequence member values.

Maple

Maplecode.

Mathematica

Wolfram Languagecode.

Program

OriginallyMapleandMathematicawere the preferred programs for calculating sequences in the OEIS, and they both have their own field labels. As of 2016,Mathematica was the most popular choice with 100,000 Mathematica programs followed by 50,000PARI/GPprograms, 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:

A016623	3, 8, 3, 9, 4, 5, 2, 3, 1, 2,...	Decimal expansion ofln(93/2).
A046543	1, 1, 1, 3, 8, 3, 10, 1, 110, 3, 406, 3	First numerator and then denominator of the central elements of the 1/3-Pascal triangle (by row).
A035292	1, 3, 8, 3, 12, 24, 16, 3, 41, 36, 24,...	Number of similar sublattices ofZ4of indexn2.
A046970	1, −3, −8, −3, −24, 24, −48, −3, −8, 72,...	Generated fromRiemann zeta function...
A058936	0, 1, 3, 8, 3, 30, 20, 144, 90, 40, 840, 504, 420, 5760, 3360, 2688, 1260	Decomposition of Stirling'sS(n,2) based on associated numeric partitions.
A002017	1, 1, 1, 0, −3, −8, −3, 56, 217, 64, −2951, −12672,...	Expansion ofexp(sinx).
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 supporting stable period-3 orbits in thelogistic map.

Keyword

The OEIS has its own standard set of mostly four-letter keywords that characterize each sequence:

• 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 specificpositional base.For example, 2, 3, 5, 7, 11, 101, 131, 151, 181...A002385are prime numbers regardless of base, but they arepalindromicspecifically in base 10. Most of them are not palindromic in binary. Some sequences rate this keyword depending on how they are defined. For example, theMersenne primes3, 7, 31, 127, 8191, 131071,...A000668does not rate "base" if defined as "primes of the form 2^n − 1". However, defined as "repunitprimes in binary, "the sequence would rate the keyword" base ".
• bref- "sequence is too short to do any analysis with", for example,A079243,the number ofisomorphism classesofassociativenon-commutativenon-anti-associativeanti-commutativeclosedbinary operationson asetof ordern.
• changedThe sequence is changed in the last two weeks.
• cofr- The sequence represents acontinued fraction,for example the continued fraction expansion ofe(A003417) or π (A001203).
• cons- The sequence is a decimal expansion of amathematical constant,likee(A001113) or π (A000796).
• core- A sequence that is of foundational importance to a branch of mathematics, such as the prime numbers (A000040), the Fibonacci sequence (A000045), etc.
• dead- This keyword used for erroneous sequences that have appeared in papers or books, or for duplicates of existing sequences. For example,A088552is the same asA000668.
• dumb- One of the more subjective keywords, for "unimportant sequences," which may or may not directly relate to mathematics, such aspopular culturereferences, arbitrary sequences from Internet puzzles, and sequences related tonumeric keypadentries.A001355,"Mix digits of pi and e" is one example of lack of importance, andA085808,"Price is Right wheel" (the sequence of numbers on theShowcase Showdownwheel used in the U.S. game showThe Price Is Right) is an example of a non-mathematics-related sequence, kept mainly for trivia purposes.
• 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,...A000027,where each term is 1 more than the previous term. The keyword "easy" is sometimes given to sequences "primes of the formf(m) "wheref(m) is an easily calculated function. (Though even iff(m) is easy to calculate for largem,it might be very difficult to determine iff(m) is prime).
• eigen- A sequence ofeigenvalues.
• fini- The sequence is finite, although it might still contain more terms than can be displayed. For example, the sequence field ofA105417shows 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 representingrational numbers.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 ofEgyptian fractions,such asA069257,where the sequence of numerators would beA000012.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 thesupersingular primesA002267,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 manyn-spherescan touch anothern-sphere of the same size? "A001116lists 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 theOEIS 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 areA331124A347347.
• more- More terms of the sequence are wanted. Readers can submit an extension.
• mult- The sequence corresponds to amultiplicative function.Terma(1) should be 1, and terma(mn) can be calculated by multiplyinga(m) bya(n) ifmandnarecoprime.For example, inA046970,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 "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.,n³,the cubes, which are all nonnegative fromn= 0 forwards) and those that by definition are completely nonnegative (e.g.,n²,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 theabsolute valuefunction.
• tabf- "An irregular (or funny-shaped) array of numbers made into a sequence by reading it row by row." For example,A071031,"Triangle read by rows giving successive states ofcellular automatongenerated 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 isPascal's triangleread by rows,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,A072036,which was presented to theInternet Oracleto ponder.
• walk- "Counts walks (orself-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...A005589,"Number of letters in the English name ofn,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. SequenceA073502,themagic constantforn×nmagic squarewith prime entries (regarding 1 as a prime) with smallest row sums, is an example of a sequence with offset 3, andA072171,"Number of stars of visual magnituden."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 thelazy caterer's sequence,the maximum number of pieces you can cut a pancake into withncuts, the OEIS gives the sequence as 1, 2, 4, 7, 11, 16, 22, 29, 37,...A000124,with offset 0, whileMathworldgives 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, namelyn= 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. ThusA000001,which starts 1, 1, 1, 2 with the first entry representinga(1) has1, 4as 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

Plot of Sloane's Gap: number of occurrences (ylog scale) of each integer (xscale) in the OEIS database

In 2009, the OEIS database was used by Philippe Guglielmetti to measure the "importance" of each integer number.The result shown in the plot on the right shows a clear "gap" between two distinct point clouds,the "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 formaⁿ(green) andhighly composite numbers(yellow). This phenomenon was studied byNicolas Gauvrit,Jean-Paul Delahayeand 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,evennumbers, geometric and Fibonacci-type sequences and so on.Sloane's gap was featured on aNumberphilevideo in 2013.

See also

• List of OEIS sequences
• Abramowitz and Stegun

References

• Borwein, J.;Corless, R. (1996)."The Encyclopedia of Integer Sequences (N. J. A. Sloane and Simon Plouffe)".SIAM Review.38(2): 333–337.doi:10.1137/1038058.
• Catchpole, H. (2004)."Exploring the number jungle online".ABC Science.Australian Broadcasting Corporation.
• Delarte, A. (November 11, 2004). "Mathematician reaches 100k milestone for online integer archive".The South End:5.
• Hayes, B.(1996)."A Question of Numbers"(PDF).American Scientist.84(1): 10–14.Bibcode:1996AmSci..84...10H.Archived fromthe original(PDF)on 2015-10-05.Retrieved2010-06-01.
• Peterson, I.(2003)."Sequence Puzzles"(PDF).Science News.163(20). Archived fromthe original(PDF)on 2017-05-10.Retrieved2016-12-24.
• Rehmeyer, J. (2010)."The Pattern Collector — Science News".Science News.www.sciencenews.org. Archived fromthe originalon 2013-10-14.Retrieved2010-08-08.

Further reading

• Roberts, S. (May 21, 2023),"What Number Comes Next? The Encyclopedia of Integer Sequences Knows.",The New York Times,retrieved21 May2023
• Sloane, N. J. A.(1999)."My favorite integer sequences"(PDF).In Ding, C.; Helleseth, T.;Niederreiter, H.(eds.).Sequences and their Applications (Proceedings of SETA '98).London: Springer-Verlag. pp. 103–130.arXiv:math/0207175.Bibcode:2002math......7175S.
• Sloane, N. J. A.(2003)."The On-Line Encyclopedia of Integer Sequences"(PDF).Notices of the American Mathematical Society.50(8): 912–915.
• Sloane, N. J. A.;Plouffe, S.(1995).The Encyclopedia of Integer Sequences.San Diego: Academic Press.ISBN0-12-558630-2.
• Zabolotskii, A. (2022)."The On-Line Encyclopedia of Integer Sequences in 2021".Mat. Pros.Series 3.8:199–212.
• Billey, Sara C.;Tenner, Bridget E. (2013)."Fingerprint databases for theorems"(PDF).Notices of the American Mathematical Society.60(8): 1034–1039.arXiv:1304.3866.Bibcode:2013arXiv1304.3866B.doi:10.1090/noti1029.S2CID14435520.

External links

• Official website
• Wikiat OEIS