The On-Line Encyclopedia of Integer Sequences

Abstract

We all recognize 0, 1, 1, 2, 3, 5, 8, 13,... but what about 1, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15,...? If you come across a number sequence and want to know if it has been studied before, there is only one place to look, the On-Line Encyclopedia of Integer Sequences (or OEIS). Now in its 49th year, the OEIS contains over 220,000 sequences and 20,000 new entries are added each year. This article will briefly describe the OEIS and its history. It will also discuss some sequences generated by recurrences that are less familiar than Fibonacci's, due to Greg Back and Mihai Caragiu, Reed Kelly, Jonathan Ayres, Dion Gijswijt, and Jan Ritsema van Eck.

Full text

arXiv:math/0312448v1 [math.CO] 24 Dec 2003
The On-Line Encyclopedia of Integer Sequences
N. J. A. Sloane
∗
1. Introduction
This article gives a brief introduction to the On-Line Encyclopedia of Integer Sequences (or
OEIS). The OEIS is a database of nearly 90, 000 sequences of integers, arranged lexicograph-
ically. The entry for a sequence lists the in itial terms (50 to 100, if available), a d escription,
formulae, programs to generate the sequence, references, links to relevant web pages, and other
information.
2. To consult the database
Since 1996, an electronic version [
20] has been accessible via the Internet, at the URL
http://www.research.att.com/∼njas/sequences/.
If a list of numbers is entered there, the reply will display the entries for all matching sequences.
For example, suppose you were trying to count the ways to insert p arentheses into a string
of n letters so that the parentheses are balanced and there are at least two letters inside each
pair of parentheses. The outer pair of parentheses is to be ignored. For n = 3 and 4 there are
respectively 3 and 11 possibilities:
n = 3 : abc, (ab)c, a(bc);
n = 4 : abcd, (ab)cd, a(bc)d, ab(cd), (ab)(cd), (abc)d,
a(bcd), ((ab)c)d, (a(bc))d, a((bc)d), a( b(cd)) .
Fur ther work shows that for n = 1, . . . , 5 the numbers are 1, 1, 3, 11, 45. Entering these into
the web page produces nine matching sequences, but they are sorted, with the most probable
match appearing first. Indeed, this entry tells you that these are the numbers (sequence
A1003)
arising from “Schr¨oder’s second problem”, and are also known as “super-Catalan numbers”.
The reply (an abridged version is shown in Figure
1) gives 21 references, ranging from
Schr¨oder (1870) [
17] to articles published electronically in the last few years. There is an
∗
Neil J. A. Sloane is with AT&T Shannon Labs, Florham Park, NJ. His email address is
njas@research.att.com.

explicit formula:
a(n) =
1
n
n−2
X
k=0

2n −k − 2
n −1

n −2
k

, n > 1 ,
a r ecur rence:
a(1) = a(2) = 1 ,
(n + 1)a(n + 1) = 3(2n −1)a(n) − (n − 2)a(n −1) , n > 1 ,
programs to produce the s equence in Maple and Mathematica, and much more.
There’s no other reference work that will carry out this kind of search.
The Encyclopedia can also be consulted via email. There are two addresses. Sending email
to
sequences@research.att.com with a line in the body of the message saying
lookup 1 1 3 11 45
will trigger the same s earch that the web page performs, only now the results are sent, almost
immediately, via email. Superseeker (
superseeker@research.att.com) carries out a more sophis-
ticated analysis and tries hard to find an explanation for the sequence, even if it is not in the
database. If the simple lookup fails, Superseeker carries out many other tests, including:
• applying over 130 transformations to the sequence, including the binomial, Euler, M¨obius,
etc. transforms [
1], and checking to see if the result is in the database
• applying Pad´e approximation metho ds to try, for example, to express the n
th
term as
a rational f unction of n (using the “gfun ” package of Salvy and Zimmermann [
16], th e
“guesss” program of Derksen [5] and the “RATE” program of Kratthentaler [8])
• checking to see if changing one or two characters produces a sequence in th e database.
Since Superseeker carries out a nontrivial amount of calculation, users are asked to submit
only one sequence per h our.
The electronic version of the Encyclopedia had its origins in the books [18] (1973) and
[21] (1995). Disk space is cheap, and the present incarnation (excluding illustrations) contains
about 72 times as much data as the 1995 book. The history of the Encyclopedia is d escribed
in more detail in [
19].
2

3. Applications
Most people us e the Encyclopedia to identify a sequence, as illustrated above. It has been
around long en ough so that there is a good chance that your sequence will be there. If not,
you will s ee a message encouraging you to submit it.
Most of these applications are unspectacular, akin to looking up a word in a dictionary (cf.
[
2]). One encounters a s equ en ce in the middle of a calculation, perhaps
1 2 4 6 10 12 16 18 22 28 30 . . . ,
and one wants to know quickly what it is — p referably a formula for the n-th term (in this
case it is probably prime(n) − 1, A6093) or a recurrence. Successful applications of this type
usually go unremarked. Some are more dramatic: there is a web page
†
that lists several
hundred articles that acknowledge help from the OEIS. One quotation will serve to illustrate
this. Emeric Deutsch of Poytechnic University, Brooklyn, said in a r ecent email message: “...
your database is invaluable. For example, for a certain sequence a
n
, usin g Maple I found the
first 100 or so indices i for which a
i
is odd. Only th e OEIS could tell me that the sequence of
these i’s is a know n sequence related to the Thue-Morse sequence. Of course, th is had to be
followed by further reading and proof.”
The other main application is to find out the cur rent status of work on a prob lem — for
example, the search for Mersenne primes (see
A43), the enumeration of Hadamard matrices
(
A7299), L atin s quares (A315) or meanders (A5316) of order n, the latest information about
the decimal expansion of π (
A796) or, better, its continued fraction expansion (A1203).
Of course people tryin g to solve puzzles or IQ tests find the database useful. A5228 is a
classic:
1 3 7 12 18 26 35 45 56 69 83 . . .
There are also s ome less obvious applications. One is in simp lifying complicated integer-
valued expressions. You might, for example, have encountered the sum
n
X
k=0

4n + 1
2n −2k

n + k
k

.
There are powerful methods for evaluating such sums [
12], [13], b ut it doesn’t take long to
work out the first few terms: 1, 12, 240, 5376, and to look them up in the database. In this
†
http://www.research.att.com/∼njas/sequences/cite.html
3

case you would have been lucky. The reply suggests that this is sequence A6588, 4
n

3n
n

,
and supplies, with references, the binomial coefficient identity you were hoping for.
Another application is in proving inequalities. You might suspect that σ(n) < n
√
n for
n > 2, where σ is the sum-of-divisors function (
A203). If the initial terms of [n
√
n] − σ(n)
(where [ ] denotes the “floor” function) are submitted to the database, the r ep ly s uggests that
this is
A55682, and points you to a reference that gives a proof of your inequality.
I cannot resist mentioning sequence
A57641, which gives the values of
[H
n
+ exp(H
n
) log(H
n
)] − σ(n)
for n ≥ 1, where H
n
is the harmonic number
P
n
i=1
1/i. Lagarias [
9], extending earlier work of
Robin [15], has shown that this sequence is non negative if and only if the Riemann hypothesis
holds!
Although the database contains a number of sequences of both of th e above typ es, I have
not made a systematic search through reference works such as [
7], [11] and it would be nice to
get many more examples.
The database can also be used to save sp ace when referring to particular sequences. When
introducing the Motzkin numbers, for example, instead of giving the definition, references and
the first few terms it is simpler just to say “. . . the Motzkin numbers M
n
(sequence
A1006 of
[20])”.
One can also search the database for sequences that mention a particular name (Riemann,
say), and there is a separate alphabetical index, useful for keeping track of all sequences on a
certain topic — e.g. the entry for groups lists abelian (
A688), primitive permutation (A19),
transitive permutation (
A2106), simple (A5180), total number (A1), . . ..
In the past year the main look-up page has been translated into 28 languages, with the goal
of making it easier to use throughout the world. The entries from the database still appear in
English, but the headings in the r ep lies and the error messages have also been translated.
4. The database
To be included in the database a sequence should be integer-valued, well-defined and interest-
ing.
The main sources are combinatorics, numb er theory and recreational mathematics, but
most branches of mathematics are represented (e.g.
A27623, the number of rin gs with n
4

elements), and there are hundreds of entries from chemistry and physics (e.g. A8253, the
co ordination sequence for d iamond : the number of carbon atoms that are n bonds away from
a particular carbon atom).
Sequences of rational numbers are entered as a linked pair giving numerators and den omi-
nators s ep arately. The Bernoulli numbers B
n
are
A27641/A27642.
Triangular arrays of numbers are read row-by-row, so that Pascal’s triangle gives A7318:
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, . . . .
Square arr ays are r ead by anti-diagonals, so that the Nim-addition table
0 1 2 3 4 5 . . .
1 0 3 2 5 4 . . .
2 3 0 1 6 7 . . .
3 2 1 0 7 6 . . .
. . . . . . . . . . . .
produces A3987:
0, 1, 1, 2, 0, 2, 3, 3, 3, 3, 4, 2, 0, 2, 4, . . . .
Most well-defined submissions get accepted, since an open-door policy seems the best. The
amazing coincidences of the Monstrous Moonshine investigations [
4], for example, m ake it
difficult to say that a particular sequence, no matter how obscure, will never be of interest.
Sequences that are discouraged are those that depend on an arbitrary an d large p arameter:
primes of the form x
2
+ y
2
+ 2003, say, whereas primes of the form x
2
+ y
2
+ 1 form a perfectly
acceptable sequence (
A79545).
The Encyclopedia currently receives between 10, 000 and 12, 000 downloads per day. The
rate of arrival of new sequences has remained constant at about 10, 000 per year for the past
seven years, with roughly the same number of comments and updates. To keep this flood of
information from getting out of control, people are asked to use a web form
‡
when submitting
new sequences or comments.
For most of its life the Encyclopedia has been maintained by the author, but in the past
year a board of associate editors has been formed to help with the work. There is also a
group of regular users who constantly sen d corrections and extensions, and help maintain the
accuracy of the entries.
‡
http://www.research.att.com/∼njas/sequences/submit.html
5

Even so, much remains to be done. There are more journals and e–print servers now than
ever, and the trained eye sees integer sequences everywhere. I still discover articles in the
library or on the web where authors h ave published sequences without sending them to the
Encyclopedia. If you come across an integer sequence in your own work or elsewhere, please
submit it to the Encyclopedia!
Of course, accuracy is a major concern in maintaining the database. The entries in [
18]
and [
21] were checked very thoroughly, and almost all th e errors that have been discovered in
those books were already present in the sour ces from which the sequences were taken . As the
number of sequences has increased in recent years, it has become more difficult to check them
all. However, the number of users h as also increased, and a large number of the entries carry
a comment that the sequence has been extended (or sometimes, corrected and extended) by
someone. Contributors see a reminder that the standards are those of a mathematics reference
work, and all submissions should be carefully checked. So, on the whole, users can be confident
that the sequences are correct. The keywords “uned” and “obsc” indicate sequences that have
not yet been edited or for wh ich the definition is unclear. These serve both as warnings to
users and to indicate places where volunteers can help.
One of the pleasures of maintaining the database is seeing the endless flow of new sequences.
I will end by mentioning a few recent examples:
Home primes (A37274), [6]: a(n) is the prime reached when you start with n, concatenate
its prime factors, and repeat until a prime is reached (a(n) is defin ed to be −1 if no prime is
ever reached, although it is conjectured that this never happens). E.g. 8 = 2 ×2 ×2 → 222 =
2 × 3 ×37 → 2337 = 3 ×19 × 41 → 31941 → . . . → (after 13 steps) 3331113965338635107,
a prime; so a(8) = 3331113965338635107:
1 2 3 211 5 23 7 3331113965338635107 311 773 11 223 13 . . . .
The EKG sequence (
A64413), [10]: a(1) = 1, a(2) = 2 and, for n ≥ 3, a(n) is th e smallest
natural number not in {a(k) : 1 ≤ k ≤ n −1} with the property that gcd{a(n −1), a(n)} ≥ 2:
1 2 4 6 3 9 12 8 10 5 15 18 14 7 21 24 16 20 22 11 . . . .
Lacing a shoe (
A78601), [14]: Number of ways to lace a shoe that has n pairs of eyelets.
The lace must follow a Hamiltonian path through the 2n eyelets, and at least one of the
neighbors of every eyelet must be on the other side of the shoe.
1 3 42 1080 51840 3758400 382838400 52733721600 . . . .
6

A “bootstrap” sequence (A79000), [3]: a(n) is taken to be the smallest positive integer
greater than a(n −1) which is consistent with the assertion “n is a member of the sequence if
and only if a(n) is o dd.”
1 4 6 7 8 9 11 13 15 16 17 18 19 20 21 23 25 27 29 31 . . . .
References
[1] M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Appl.,
226–228 (1995), 57–72; Erratum 320 (2000), 210. [arXiv: m ath.CO /0205301]
[2] B. Cip ra, Mathematicians get an on-line fingerprint file, Science, 265 (22 July, 1994), 473.
[3] B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson’s
sequence, J. Integer Sequences, 6 (2003), #03.2.2. [arXiv: math.NT/0305308]
[4] J. H. Conway and S . P. Norton, Monstrous moonshine, Bull. London Math. Soc., 11
(1979), 308–339.
[5] H. Derken,
An algorithm to compute generalized Pad´e-Hermite forms, Report 9403, De-
partment of Mathematics, Catholic Un iversity Nijmegan, 1994.
[6] P. De Geest,
Home primes < 100 and beyond, published electronically at
www.worldofnumbers.com/topic1.htm, 2003.
[7] H. W. Gould, Combinatorial Identities, Morgantown, WV, 1972.
[8] C. Krattenthaler, RATE-A Mathematica guessing machine, available electronically from
http://euler.univ-lyon1.fr/home/kratt/rate/rate.html.
[9] J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Amer.
Math. Monthly, 109 (2002), 534–543.
[10] J. C . Lagarias, E. M. Rains and N. J. A. Sloane,
The EKG sequence, Experimental Math.,
11 (2002), 437–446. [arXiv: math.NT/0204011]
[11] D. S. Mitrinovi´c, J. S´andor and B. Crstici, Handbook of Number Theory, Kluwer, Dor-
drecht, 1996.
7

[12] I. Nemes, M. Petkovˇsek, H. S. Wilf and D. Zeilberger, How to do Monthly problems with
your computer, Amer. Math. Monthly, 104 (1997), 505–519.
[13] M. Petkovˇsek, H. S. Wilf and D. Zeilberger,
A = B, Peters, Wellesley, MA, 1996.
[14] B. Polster, What is the best way to lace your s hoes?, Nature, 420 (Dec. 5, 2002), 476.
[15] G. Rob in, Grandes valeurs de la fonction somme des diviseurs et hyp oth`ese de Riemann
[Large values of the sum-of-divisors function and th e Riemann hypothesis], J. Math. Pures
Appl., 63 (1984), 187–213.
[16] B. Salvy and P. Zimmermann,
Gfun: a Maple package for the manipulation of generating
and holonomic functions in one variable, ACM Transactions on Mathematical Software,
20 (1994), 163–177.
[17] E. S chr¨oder, Vier combinatorische Probleme [Four combinatorial problems], Zeit. f. Math.
Phys., 15 (1870), 361–376.
[18] N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, NY, 1973.
[19] N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Pro-
ceedings of SETA ’98), edited by C. Ding, T. Helleseth and H. Niederreiter, Springer–
Verlag, London, 1999, pp. 103–130. [arXiv: math.CO/0207175]
[20] N. J. A. Sloane,
The On-Line Encyclopedia of Integer Sequences, published electronically
at http://www.research.att.com/∼njas/sequences/, 2003.
[21] N. J. A. Sloane and S. Plouffe,
The Encyclopedia of Integer Sequences, Academic Press,
1995.
8

Figure 1: Part of the reply when the s equence 1, 1, 3, 11, 45 is submitted to the On-Line
Encyclopedia. Many references, links and comments have been omitted to save sp ace.

Greetingsfromthe
On-LineEncyclopediaofIntegerSequences!
Matches(uptoalimitof30)foundfor1131145:
IDNumber:A001003
Sequence:1,1,3,11,45,197,903,4279,20793,103049,518859,2646723,
13648869,71039373,372693519,1968801519,10463578353,
55909013009,300159426963,1618362158587,8759309660445,
47574827600981,259215937709463,1416461675464871
Name:
Schroeder'ssecondproblem(generalizedparentheses);alsocalled
super-CatalannumbersorlittleSchroedernumbers.
Comments:a(n)=numberofwaystoinsertparenthesesinastringofnsymbols.The
parenthesesmustbebalancedbutthereisnorestrictiononthenumber
ofpairsofparentheses.Thenumberoflettersinsideapairof

parenthesesmustbeatleast2.Parenthesesenclosingthewholestring
areignored.
Stanleygivesseveralotherinterpretationsforthesenumbers.
ReferencesP.J.Cameron,Somesequencesofintegers,DiscreteMath.,75(1989),
89-102.
I.M.H.Etherington,Someproblemsofnon-associativecombinations,
EdinburghMath.Notes,32(1940),1-6.
T.S.Motzkin,Relationsbetweenhypersurfacecrossratios,anda
combinatorialformulaforpartitionsofapolygon,forpermanent
preponderance,andfornon-associativeproducts,Bull.Amer.Math.
Soc.,54(1948),352-360.
E.Schroeder,ViercombinatorischeProbleme,Zeit.f.Math.Phys.,15

(1870),361-376.
R.P.Stanley,EnumerativeCombinatorics,Cambridge,Vol.2,1999;
seepage178.
.......................................
Links:H.Bottomley,
Illustrationofinitialterms
E.PergolaandR.A.Sulanke,

SchroederTriangles,Paths,andParallelogramPolyominoes,J.Integer
Sequences,1(1998),#98.1.7.
Formula:Recurrence:a(1)=a(2)=1;forn>1,(n+1)a(n+1)=
3(2n-1)a(n)-(n-2)a(n-1).G.f.:(1/4)*(1+x-sqrt(1-6*x+x^2)).
Forn>1,a(n)=(1/n)*sum_{k=0..n-2}
binomial(2*n-k-2,n-1)*binomial(n-2,k).
Example:
a(3)=3:abc,a(bc),(ab)c;a(4)=11:abcd,(ab)cd,a(bc)d,ab(cd),
(ab)(cd),a(bcd),a(b(cd)),a((bc)d),(abc)d,(a(bc))d,((ab)c)d.
Maple:
t1:=(1/4)*(1+x-sqrt(1-6*x+x^2));series(t1,x,40);
Math'ca:
Sch[1]=Sch[2]=1;Sch[n_Integer]:=Sch[n]=(3(2n-3)Sch[n-1]
-(n-3)Sch[n-2])/n;Array[Sch[#]&,20]
Seealso:See
A000108,A001190,A001699,A000081forotherwaystocount
parentheses.Cf.
A000311,A010683,A065096.
Keywords:nonn,easy,nice

1of1 1/31/20031:21PM
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