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# On-Line Encyclopedia of Integer Sequences

## From Wikipedia, the free encyclopedia

Created by Neil Sloane oeis.org 45,356 (August 2019)[1] No[2] Optional[3] 1996; 23 years ago

The On-Line Encyclopedia of Integer Sequences (OEIS), also cited simply as Sloane's, is an online database of integer sequences. It was created and maintained by Neil Sloane while a researcher at AT&T Labs. He transfered the intellectual property and hosting of the OEIS to the OEIS Foundation in 2009.[4] Sloane is president of the OEIS Foundation.

OEIS records information on integer sequences of interest to both professional and amateur mathematicians, and is widely cited. As of June 2019 it contains over 320,000 sequences, making it the largest database of its kind.

Each entry contains the leading terms of the sequence, keywords, mathematical motivations, literature links, and more, including the option to generate a graph or play a musical representation of the sequence. The database is searchable by keyword and by subsequence.

### YouTube Encyclopedic

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• ✪ ENCODE Project Update - Elise Feingold
• ✪ UW Allen School Colloquium: Hannaneh Hajishirzi (UW)
• ✪ Possible three letter words | Probability and Statistics | Khan Academy
• ✪ Georg Cantor

#### Transcription

Elise Feingold: Thank you, Rudy. So since it’s been about two years -- a little more than two years since we’ve given you an update on the ENCODE project, I thought I would start with a little bit of background and motivation for the project as a context for this talk. So back in 2003, when we were anticipating the completion of the human genome sequence, we were asking the question, “How can we read the sequence?” We knew we had the genetic code, we were moderately good at identifying protein coding regions, but bind structures were difficult to predict from the sequence. We had no powerful regulatory code. We knew that evolutionary conservation can help to identify functioning important regions. At that time, it was estimated to be about five percent of the genome was conserved, one and a half percent of the genome was protein coding. We were really interested in finding out what the function of the noncoding conserved sequences are, as well as the function of the non-conserved sequences. Also over the last number of years, we have an increasing understanding that noncoding DNA is important for disease and evolution. We know now that about 90 percent of GUS findings lie outside of protein coding regions. We know that noncoding DNA variance can cause human diseases and older human traits, such as in the fragile lex syndrome and ALS. And also that 80 percent of recent adaptation signatures lay outside of protein coding regions. So back in 2003, we starting the ENCODE project. It stands for Encyclopedia of DNA Elements. And the goal is to compile a comprehensive encyclopedia of all sequence features in the human genome and in the genomes of select model organisms. One of the hallmarks of this project is rapid prepublication data release. We really wanted to create a resource that was freely available to the community to enhance our understanding of gene regulation, as well as the genetic basis of disease. This slide is a cartoon of the various data types that are being generated by the ENCODE resource. I don’t have time to go into this in detail, but I just want to point out that the consortium uses a number of hythroupa [spelled phonetically] technologies that are primarily sequenced based, including for example, chip seek and RNA seek, to measure a number of biochemical properties of the genome, including a mapping of transcription factor binding sites, histone modifications, demethylation and RNA transcription. And all of these data are then mapped into the genome to map transcripts, gene promoters, putative regulatory regions, such as enhancers. This slide outlines the timeline for the ENCODE project. I mentioned that we started back in 2003. We started with a pilot project, with a well-defined one percent of the human genome. And at the same time we initiated a number of technology development initiatives, which we felt was going to be important in order to create the resource we were interested in making. Based on the success of the pilot project, in 2007, we launched the production phase of ENCODE, which -- where we went from one percent of the genome to 100 percent and complete genome analysis in the human genome. At the same time, in 2007, we launched the modENCODE Consortium, which focused on creating a catalog of functional elements in the worm and fly genomes. And then in 2009, with the availability of some economic stimulus money, the R money, we launched a small effort in the mouse ENCODE. These projects ended in 2012, and when we started ENCODE phase three, which we are about half way through now, we focused on expanding the catalog in the human and mouse genomes and we added in some computational analysis program, as well. We felt that it was important to expand the expertise and computational analysis in order to best make use of the ENCODE resource. And at that time we funded the fourth technology development initiative as well. So, Eric mentioned this this morning, it seems like a couple of days ago already, so I just want to remind everyone about the modENCODE Publication. So modENCODE -- funding for that program ended in 2012, and this summer there was a series of publications that kind of the capstone of the project. This included worm, fly, and human comparative analyses. There were three man integrative papers published in Nature at the end of August. These were focused on transcription, chromatin, and regulation. And these papers found that there were commonalities in these processes across these three evolutionary, quite diverse, organisms. In addition, there were about at least 15 companion papers that were published in ‘Nature,’ ‘Genome Research,’ ‘Genome Biology’ and other journals -- that was this summer -- and those join another approximately 150 publications that the modENCODE Consortium had already published. And there was an additional 150 publications or so that was published by researchers outside of the consortium but they were using the modENCODE data. We have compiled al of the information about these modENCODE publications, at the ENCODE portal at Stamford, the www.edcodeproject.org/comparative, where you can get links directly to the data as well as all of these publications and hopefully that will make it easier for the research community. Okay. So I just want to expand a little bit on the goals of ENCODE three, which is, we’re in the current phase of ENCODE three phase. We wanted to expand toward completion the catalog of functional elements in the human and mouse genomes. We added additional cell types, including more primary sources, additional data types, such as RNA binding proteins, which had not been included in ENCODE two. We’ll continue high quality data using hythro [spelled phonetically] pipelines. We’re developing new technologies analytical tools to generate, analyze and validate the data, and we’re providing the data and tools to the community in useful forms as possible. We're wanting to provide easy and rapid access. A key thing is to provide sufficient meta data, so people understand how the experiment were performed, and we want to provide various ways to use it, and also recognize the diverse needs of different types of users. This slide outlines the structure of the ENCODE III. There are seven data production groups. They deposit their data into the data coronation center, which performs quality assessments. They house the data and they just make the data available to the research community. The data is taken out by the data analysis center, which supports the activities of the analysis working groups. This is where our computational analysis groups are involved. And the output are gene models, chromatin states, and candidate functional elements, which, in total, represent the encyclopedia. I think Eric also mentioned this this morning, but one of the features of the current phase of ENCODE, is we have a revised data release policy. The key point is here, where external data users may freely download, analyze, and publish results based on any ENCODE data without restriction as soon as the data are released. We just ask that the consortium is acknowledged, as well as the individual data producers, and the specific accession numbers. Additional ENCODE three features include cloud computing. We heard a little bit about that this morning. The data is available at Amazon Web Services, or AWS. We are creating uniform processing pipelines that will be available at DNANexus, and this provides transparency about having code data as processed. And these pipelines are available for the community to use as their own data, and will be continually updated as the pipelines are updated. We’ve taken the lead on data interoperability. We’re coordinating ontology selection and metadata standards with related projects. And I just want to say a word about the genomic sharing policy that was mentioned earlier today. In addition to rapid prepublication data release, ENCODE is continuing to work ahead of this policy. If you recall, it’s not going to be in place until January of 2015, and that’s for new projects. But we’ve been working to develop sample consent language for open access for genomic data, and we’re currently working to obtain a wide range of samples using the new contestants. Okay, there is a new ENCODE portal at www.ENCODEproject.org, which is hosted at -- by Stanford University, the Data Coordination Center. We hope that the community will find enhancements that are made on this portal useful in accessing the data. I just want to point out a couple of features. If you click on the data tab, you can get access to the essays, the bio samples, and the antibodies, and the characterization of the antibodies. And there’s metadata driven searches that could really help one narrow down to get to the data types of interest. I've shown one example here where you can click on different tabs on the left side of the panel here if you’re interested specifically. If [unintelligible] interested in chip seq data, if you’re interested exclusively in mouse data, you can click on that, different bio samples, types, if you want to use primary tissues. And then just looking specifically at the organs, you can actually click on multiple ones. Here I clicked on liver and kidney, and then looking at the life stage, you’re looking specifically for embryonic tissue, and here it shows 28 samples -- data type, data sets that fit this criteria and this will hopefully enable the researchers to find what they’re looking for. ENCODE has developed a number of data standards and software tools. If you click under “Methods,” you can find a lot of this information. There are experimental guidelines, including data standards for many of the common assays that are used in code. We can elaborate information about quality metrics, as well as software tools that were used to create the resource. This includes tools that were created by ENCODE, or used by other groups for the resource. We also monitor and code publications as mentioned earlier, with modENCODE, and Eric has mentioned earlier. This is one measure of the impact of the resource, and if you click on publications under modENCODE, you can get information about previous ENCODE integrative analyses. You’re all probably aware that in 2012, ENCODE published about 30 papers that were integrated in a publication package. You can get the details of those here, you can look at other ENCODE-funded publications, those published by the ENCODE consortium, by the modENCODE consortium, and the modENCODE consortium, but we’re also trying to track community publications, which is a little tricky, since it’s really hard to search on ENCODE to find all publications, but we have narrowed down where we’re finding publications that actually use ENCODE data. Eric showed this slide earlier. This shows, in purple, the number of publications by the ENCODE consortium. There are currently about 360 publications by ENCODE investigators. And in the blue, the number of community publications at the current time is about 660 publications. So in total, about 1,000 publications, but about two-thirds of these are coming from the research community, and so we’re very excited to see this increase over time. We categorized these by what researchers, the outside -- sorry, the community publications, by which researchers are using them for. We have about 160 publications that are disease-specific, and we’ve analyzed, categorized what diseases they are studying. Not surprisingly, about a third of the publications are cancer research, 15 percent looking at autoimmunity and allergy, 13 percent in neurology and psychiatric diseases, and about seven percent in cardiovascular disease. But there’s also a wide range of other disorders that are -- have been published using ENCODE data in smaller percentages. Okay, just want to mention very briefly some outreach activities. ENCODE has had a series of tutorials. Many of these have been done in conjunction with the robot epigenomics program. We’ve held these tutorials at ASHG in 2012, 2013, and another one already sold out is planned for this year’s meeting in San Diego. And we’ve had a tutorial also about the biology of genomes, and also all the materials from these tutorials, the handouts are available on genome.gov. We held a workshop -- ENCODE charge consortium workshop in January. This is a consortium focused on heart and aging-related research. And we had a one-day workshop on how to use the data, and we formalized collaborations with this group. But we’d like to expand our impact, and so we’re planning to have a user’s meeting in 2015 to bring, you know, much wider swath of the research community to get to know the -- [end of transcript]

## History

Neil Sloane started collecting integer sequences as a graduate student in 1965 to support his work in combinatorics.[5] The database was at first stored on punched cards. He published selections from the database in book form twice:

1. A Handbook of Integer Sequences (1973, ISBN 0-12-648550-X), containing 2,372 sequences in lexicographic order and assigned numbers from 1 to 2372.
2. 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.

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 an e-mail service (August 1994), and soon after as a web site (1996). As a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998.[6] 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.[7] In 2004, Sloane celebrated the addition of the 100,000th sequence to the database, 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 at OEIS.org was created to simplify the collaboration of the OEIS editors and contributors.[8] 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,[9][10] following a proposal by OEIS Editor-in-Chief Charles Greathouse to choose a special sequence for A200000.[11]

## Non-integers

Besides integer sequences, the OEIS also catalogs sequences of fractions, the digits of transcendental numbers, complex numbers and so on by transforming them into integer sequences. Sequences of rationals 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, ${\displaystyle \textstyle {1 \over 5},{1 \over 4},{1 \over 3},{2 \over 5},{1 \over 2},{3 \over 5},{2 \over 3},{3 \over 4},{4 \over 5}}$, 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). Important irrational numbers 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, ... (A000796)), 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, ... (A004601)), or 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, ... (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 functions, n for running variables, etc.). 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., a(n) represents the nth term of the sequence.

### Special meaning of zero

Zero is often used to represent non-existent sequence elements. For example, A104157 enumerates the "smallest prime of n² 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 Nφ(m) (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. Occasionally −1 is used for this purpose instead, as in A094076.

### Lexicographical ordering

The OEIS maintains the lexicographical order of the sequences, so each sequence has a predecessor and a successor (its "context").[12] OEIS normalizes the sequences for lexicographical ordering, (usually) ignoring all initial zeros and ones, and also the sign of each element. Sequences of weight distribution codes often omit periodically recurring zeros.

For example, consider: the prime numbers, the palindromic primes, the Fibonacci sequence, the lazy caterer's sequence, and the coefficients in the series expansion of ${\displaystyle \textstyle {{\zeta (n+2)} \over {\zeta (n)}}}$. 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.[13] One of the earliest self-referential sequences Sloane accepted into the OEIS was A031135 (later A091967) "a(n) = n-th term of sequence An or -1 if An has fewer than n terms". This sequence spurred progress on finding more terms of A000022. A100544 lists the first term given in sequence An, but it needs to be updated from time to time because of changing opinions on offsets. Listing instead term a(1) of sequence An might seem a good alternative if it weren't for the fact that some sequences have offsets of 2 and greater. This line of thought leads to the question "Does sequence An contain the number n ?" and the sequences A053873, "Numbers n such that OEIS sequence An contains n", and A053169, "n is in this sequence if and only if n is not in sequence An". Thus, the composite number 2808 is in A053873 because A002808 is the sequence of composite numbers, while the non-prime 40 is in A053169 because it's not in 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 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.

## An abridged example of a typical OEIS entry

This entry, A046970, was chosen because it contains every field an OEIS entry can have.[14]

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)/(4pi^2))*z(n+2)/z(n) = -B(n)*((n+2)*(n+1)/(4pi^2))*Sum(j=1, infinity) [ a(j)/j^(n+2) ]
...
REFERENCES  M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, 1965, pp. 805-811.
LINKS       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].
Wikipedia, Riemann zeta function.
FORMULA     Multiplicative with a(p^e) = 1-p^2. a(n) = Sum_{d|n} mu(d)*d^2.
a(n) = product[p prime divides n, p^2-1] (gives unsigned version) [From Jon Perry (jonperrydc(AT)btinternet.com), Aug 24 2010]
EXAMPLE     a(3) = -8 because the divisors of 3 are {1, 3} and mu(1)*1^2 + mu(3)*3^2 = -8.
...
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*(x[[i]][[1]]^2 - 1)]; p}, {n, 1, 50, 1}]] [From Jon Perry (jonperrydc(AT)btinternet.com), Aug 24 2010]
PROG 	    (PARI) A046970(n)=sumdiv(n, d, d^2*moebius(d)) (Benoit Cloitre)
CROSSREFS   Cf. A027641 and A027642.
Sequence in context: A035292 A144457 A146975 * A058936 A002017 A118582
Adjacent sequences:  A046967 A046968 A046969 * A046971 A046972 A046973
KEYWORD     sign,mult
AUTHOR      Douglas Stoll, dougstoll(AT)email.msn.com
EXTENSIONS  Corrected and extended by Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 25 2001
Additional comments from Wilfredo Lopez (chakotay147138274(AT)yahoo.com), Jul 01 2005


### 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 show, the rough correspondence holds.
A059097 A060001 Numbers n such that the binomial coefficient C(2n, n) is not divisible by the square of an odd prime. Jan 1, 2001 Fibonacci(n)!. Mar 14, 2001 Number of 3-dimensional polyominoes (or polycubes) with n cells and symmetry group of order exactly 24. Jan 1, 2002 Smallest number such that n·a(n) is a concatenation of n consecutive integers ... Aug 31, 2002 Continued fraction for ζ(3/2) Jan 1, 2003 Number of permutations satisfying −k ≤ p(i) − i ≤ r and p(i) − i Feb 10, 2003 Length of longest contiguous block of 1s in binary expansion of nth prime. Nov 20, 2003 Exponential convolution of A069321(n) with itself, where we set A069321(0) = 0. Jan 1, 2004 Marks from the 22000-year-old Ishango bone from the Congo. Nov 7, 2004 Column 1 of triangle A102230, and equals the convolution of A032349 with A032349 shift right. Jan 1, 2005 Number of consecutive integers starting with n needed to sum to a Niven number. Jul 8, 2005 Triangle-free positive integers. Jan 12, 2006 Möbius transform of sum of prime factors of n with multiplicity. 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, or at least about four lines' worth. The sequence field makes no distinction between sequences that are finite but still too long to display and sequences that are infinite. To help make that determination, you need to look at the keywords field for "fini", "full", or "more". 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, (A000578) is named "The 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 triangles resulting from criss-crossing cevians within a triangle so that two of its sides are each n-partitioned," while Neil Sloane points out the unexpected relationship between centered hexagonal numbers (A003215) and second Bessel 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 to graph theory
5. others related to computer codes, more extensive tabulations in specific research areas provided by individuals or research groups
Formula
Formulae, recurrences, generating functions, etc. for the sequence.
Example
Some examples of sequence member values.
Maple
Maple code.
Mathematica
Wolfram Language code.
Program
Originally Maple and Mathematica were 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 50,000 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.
See also
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 A046543 3, 8, 3, 9, 4, 5, 2, 3, 1, 2, ... Decimal expansion of ln(93/2). 1, 1, 1, 3, 8, 3, 10, 1, 110, 3, 406, 3 First numerator and then denominator of the centralelements of the 1/3-Pascal triangle (by row). 1, 3, 8, 3, 12, 24, 16, 3, 41, 36, 24, ... Number of similar sublattices of Z4 of index n2. 1, −3, −8, −3, −24, 24, −48, −3, −8, 72, ... Generated from Riemann zeta function... 0, 1, 3, 8, 3, 30, 20, 144, 90, 40, 840,504, 420, 5760, 3360, 2688, 1260 Decomposition of Stirling's S(n, 2) based onassociated numeric partitions. 1, 1, 1, 0, −3, −8, −3, 56, 217, 64, −2951, −12672, ... Expansion of exp(sin x). 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-valuessupporting stable period-3 orbits in the logistic equation.
Keyword
The OEIS has its own standard set of mostly four-letter keywords that characterize each sequence:[15]
• base The results of the calculation depend on a specific positional base. For example, 2, 3, 5, 7, 11, 101, 131, 151, 181 ... A002385 are prime numbers regardless of base, but they are 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 primes 3, 7, 31, 127, 8191, 131071, ... 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, A079243, Number of isomorphism classes of associative non-commutative non-anti-associative anti-commutative closed binary operations on a set of order n.
• cofr The sequence represents a continued fraction, for example, continued fraction expansion of e (A003417) or π (A001203).
• cons The sequence is a decimal expansion of a mathematical constant, like e (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, A088552 is the same as 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. A001355, "Mix digits of pi and e." is one example of lack of importance, and 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) is an example of a non-mathematics-related sequence, kept mainly for trivia purposes.[16]
• 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 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 eigenvalues.
• fini The sequence is finite, although it might still contain more terms than can be displayed. For example, the sequence field of 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 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 of Egyptian fractions, such as A069257, where the sequence of numerators would be 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 primes 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-spheres can touch another n-sphere of the same size?" A001116 lists the first ten known solutions.
• hear A sequence with a graph audio deemed to be "particularly interesting and/or beautiful".
• less A "less interesting sequence".
• look A sequence with a graph visual deemed to be "particularly interesting and/or beautiful".
• 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 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 "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 positive from n = 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.
• 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, 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, 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 the Internet Oracle to ponder.
• walk "Counts walks (or 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 ... 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 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 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, ... 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 don't 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 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, with the @ character replaced by "(AT)" with some exceptions such as for associate editors or if an e-mail address does not exist. 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, followed by date of extension.

## Applications

### Sloane's gap

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.[17] The result shown in the plot on the right shows a clear "gap" between two distinct point clouds[18] 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 form an (green) and highly composite numbers (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, even numbers, geometric and Fibonacci-type sequences and so on.[19] Sloane's gap was featured on a Numberphile video in 2013.[20]

## Notes

1. ^ "Oeis.org Traffic, Demographics and Competitors - Alexa". www.alexa.com. Retrieved 7 August 2019.
2. ^ "Goals of The OEIS Foundation Inc". The OEIS Foundation Inc. Retrieved 2017-11-06.
3. ^ Registration is required for editing entries or submitting new entries to the database
4. ^ Transfer of IP in OEIS to the OEIS Foundation Inc.
5. ^ Gleick, James (January 27, 1987). "In a 'random world,' he collects patterns". The New York Times. p. C1.
6. ^
7. ^ "Editorial Board". On-Line Encyclopedia of Integer Sequences.
8. ^ Neil Sloane (2010-11-17). "New version of OEIS".
9. ^ Neil J. A. Sloane (2011-11-14). "[seqfan] A200000". SeqFan mailing list. Retrieved 2011-11-22.
10. ^ Neil J. A. Sloane (2011-11-22). "[seqfan] A200000 chosen". SeqFan mailing list. Retrieved 2011-11-22.
11. ^ "Suggested Projects". OEIS wiki. Retrieved 2011-11-22.
12. ^ "Welcome: Arrangement of the Sequences in Database". OEIS Wiki. Retrieved 2016-05-05.
13. ^ Sloane, N. J. A. "My favorite integer sequences" (PDF). p. 10.
14. ^
15. ^ "Explanation of Terms Used in Reply From". On-Line Encyclopedia of Integer Sequences.
16. ^ 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."
17. ^ Guglielmetti, Philippe. "Chasse aux nombres acratopèges". Pourquoi Comment Combien (in French).
18. ^ Guglielmetti, Philippe. "La minéralisation des nombres". Pourquoi Comment Combien (in French). Retrieved 25 December 2016.
19. ^ Gauvrit, Nicolas; Delahaye, Jean-Paul; Zenil, Hector (2011). "Sloane's Gap. Mathematical and Social Factors Explain the Distribution of Numbers in the OEIS". arXiv:1101.4470. Bibcode:2011arXiv1101.4470G. doi:10.5642/jhummath.201301.03. Cite journal requires |journal= (help)
20. ^ "Sloane's Gap" (video). Numberphile. 2013-10-15. With Dr. James Grime, University of Nottingham

## Further reading

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