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The On-Line Encyclopedia of Integer Sequences today

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This is a translation of my article The On-Line Encyclopedia of Integer Sequences in 2021, published in Mat. Pros. Ser. 3 28, 199–212 (2021). My other texts about the OEIS in Russian include:

Introduction: what does the On-Line Encyclopedia contain

The Fibonacci numbers:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610...

The sequence of the factorials of the numbers n = 0, 1, 2, 3...:

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880...

The number of partitions of the number n = 1, 2, 3... into positive integers:

1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176...

All these are integer sequences. You can meet them all around combinatorics, number theory, and recreational mathematics. And if there is a multitude of objects of the similar form, then one can create an index for these objects. The On-Line Encyclopedia of Integer Sequences, OEIS, is such an index. You can access it at oeis.org.

References to the OEIS are frequently encountered in Mat. Pros. However, the On-Line Encyclopedia inclusion criteria, its editorial process, its role in mathematics and its future are rarely discussed in the literature. Thus I hope that an article covering these aspects would be interesting for readers.

What does the OEIS contain? Sometimes it is said that the main criterion is: a sequence can make it into the OEIS if anyone ever becomes interested in it independently of its author. In other words, the sequence should be interesting.

Being interesting is, of course, a subjective concept. However the former vice president of the OEIS Foundations Charles Greathouse has written an essay Is this sequence interesting containing a classification of sequences that allows to understand how the On-Line Encyclopedia editors evaluate whether a sequence is interesting and appropriate. Largely following Greathouse, we divide sequences into the following categories.

  • Interesting and probably should be in the OEIS. A sequence generated by a simple and natural formula or an algorithm which has some unexpected properties (or at least a nice graph); a sequence originating from apparently unrelated formulas or algorithms; a sequence illustrating a nice theorem; a sequence from a known problem or puzzle. This category also includes sequences from published scientific literature.

  • Interesting but probably not for the OEIS. Very short sequences; sequences with terms which are known only approximately; sequences almost identical to the ones already in the OEIS.

  • Not interesting but possibly should be in the OEIS. Very simple sequences, such as the all-1's sequence; also sequences from literature that contain errors.

  • Not interesting and probably not for the OEIS. Sequences with definitions that are not natural.
    (Manipulations with decimal digits and prime numbers are abused most frequently – like "Prime terms of the sequence A000055", which is pointless, taking into account that A000055 counts trees on n vertices.)

Every OEIS sequence has a six-digit number with the letter A prepended; in particular, the sequences mentioned above bear numbers A000045, A000142, A000041. Every sequence entry contains comments about its properties, formulas, programs that generated it, references for the relevant literature and web pages, a text file with a long list of the terms (the so-called b-file). OEIS already has a third of a million entries and it is updated continuously.

History and organization

The founder of the Encyclopedia, Neil James Alexander Sloane, has told its story many times. In 1964, when he was a student, Sloane began collecting integer sequences. Specimens were continuously added to that collection, it has been printed in book form twice, and became relatively well-known. In 1996 the collection had as much as ten thousand sequences, and Sloane made it directly available via the Internet; that is when the Encyclopedia became On-Line. This increased both the audience and the stream of messages with comments and new sequences. Sloane had to process more than a hundred emails daily. In 2009, Sloane incorporated a non-profit OEIS Foundation and made it the owner of the OEIS content. At the same time, an attempt has been made to turn the OEIS into a wiki on the MediaWiki engine, but its built-in search engine turned out to be not suitable for sequences. Eventually, Russ Cox, a software developer from Google, wrote software (a wiki engine and search) specially for the OEIS, which is being used since 2010.

(Wiki is a website edited by users together. Software under the hood of a wiki is called a wiki engine. The most widely known wiki is the Wikipedia, which works on the MediaWiki engine.
Hereinafter the italic font is used for notes added in proof of the original text or in translation.)

For the last ten years, the OEIS works as a pre-moderated wiki. This means that anybody can sign up and suggest new sequences or edits to existing sequence entries, but these edits will be published only when editors approve them. I tell more about this process below.

Neil Sloane, who is now (when this text was originally published) 81, took the position of the President of the OEIS Foundation. The President together with the Foundation's Board of Trustees takes care of the paperwork and financial aspects. The Foundation is funded by private donations (largely coming from the OEIS editors) and the Simons Foundation grants. The OEIS Foundation spends 20–30 thousands dollars per year. The money is paid, in particular, for the server on which the OEIS is run.

In addition to the OEIS Foundation paperwork, Sloane continues investigating interesting sequences and corresponding about it. Also, he often makes the final decision in various questions related to the OEIS when editors disagree with each other.

How the OEIS works

Technologically, the OEIS consists of two parts.

The first part is the main OEIS website (oeis.org). It is an interface for the search over the OEIS database and the wiki engine for editing and extending it (created by Russ Cox in 2010). Also, the main website contains old help pages of varying quality.

The second part is the so-called OEIS Wiki. Its URL is oeis.org/wiki, and it is the MediaWiki wiki which has been set up in 2009 during the attempt to move all OEIS activity to it. This part of the OEIS contains:

  • new account creation;

  • help pages for contributors;

  • communication with other contributors;

  • the OEIS index (quite selective, mostly composed manually);

  • the list of the works referencing the OEIS, maintained manually;

  • lists of editors and the block log;

  • mathematical-ish pages, written by various authors. All OEIS content and almost all user interface is in English.

New sequences and edits to existing entries are processed at the main website. Anyone can request a contributor's account. The moderation is done by editors. The OEIS has two kinds of editors: Editors-in-Chief can publish edits and new sequences, while Associate Editors can only mark them as "reviewed", so their role is purely advisory.

(In the absence of better alternatives, my translation of these terms into Russian literally mean "Senior Editor" and "Junior Editor".)

At the time of the writing, the OIES had 35 Editors-in-Chief (currently there are more), many of whom work actively in the OEIS, and about a hundred of Associate Editors, of whom less than a dozen perform editorial work. All editors, as well as the OEIS Foundation leadership, work for free as volunteers. An OEIS contributor can become an editor by offering their service or by being granted with this status unexpectedly by Neil Sloane if he finds that the contributor understands the spirit of the On-Line Encyclopedia well enough.

Assume that a contributor prepared a draft of a new sequence entry or an edit to an existing one. Both cases are processed by editors in more or less the same way. The term "draft" is used in both cases. After creating a draft, the next step for the contributor is to mark it as ready for review by editors (the "proposed" status) by pressing the corresponding button. At any stage, the author of the draft or another user can make edits, after which it needs to be proposed for review again. The OEIS editors may leave instructions, questions or notes for other editors in the so-called "pink-box comments" to the draft, or make their own edits.

When an editor believes that the new sequence is ready for publishing, they make its status "reviewed". Then, an Editor-in-Chief makes it "approved", and at that point the sequence will be published in the OEIS. Alternatively, if the editors decide that the proposed sequence is not appropriate for the OEIS (for example, if it turns out that it is a duplicate of an existing sequence), it will be deleted, or, to be more precise, "recycled" – its content is deleted, and the number is re-used by someone else later. Approval and rejection of the edits to the existing sequence is done in the same way, only without recycling in the case of the negative outcome.

The time that a draft takes to be reviewed (or sometimes approved right away) is hard to predict. It can take literally from one hour to several months. In most cases of new sequences, it takes from a month to two months.

Usually there are about three hundred drafts at any given time. Editors themselves decide which of them to process. They have basically no substantial guidelines. There exist two private mailing lists, but they are used quite a little, so almost all communication between editors takes place in the OEIS itself, in the pink-box comments.

If you tried to become an OEIS contributor and your sequence has been published, it's great! If it has been rejected, it's OK: the OEIS is not the only platform for publishing mathematical ideas, and often not the most appropriate one.

The content of the OEIS is published on the conditions of the CC BY-NC 3.0 license, which allows using and modifying it for non-commercial goals on the condition of providing a link to the source.

Community and significance

The OEIS community is oddly attractive. People from all around the world take part in creating the On-Line Encyclopedia, and since everybody uses their real name, a wide diversity of names can be seen, including English, Russian, German, Spanish, Romanian, Chinese, Indian, Vietnamese, and so on, and so forth.

Some contributors are mathematicians (undergraduate and graduate students, emeritus professors, much more rarely – postdocs and professors, including well-known ones – e. g., I edited contributions from Donald Knuth and from Alan Sokal), some are more or less qualified amateurs: physicists, engineers, IT engineers, maths teachers, school students. OEIS might be an appropriate place to publish results of a school project if its supervisor knows the requirements for the content and the formatting of sequence entries. For example, sequences A275302A275305 (related to Langton's ant on the triangular tiling) and A275667 (related to a Life-like cellular automaton on the triangular tiling in which every pattern self-replicates) were obtained by students at the Slon summer school during a cellular automata project supervised by me.

Most authors contribute to the OEIS once or rarely. Some get deeply involved, and the OEIS becomes their years-long hobby. Regular contributors often have specialization. For example, Sean Irvine, a genomics software developer from New Zealand, works with the Encyclopedia from the moment it went online; starting in 2009, Sean constantly adds Java programs to sequence entries, beginning from A000001 "Number of groups of order n". In passing he finds and corrects errors in the OEIS; he says that it quickly became the main impetus for his work. Sean's programs are all included into a framework called jOEIS. The number of sequences which jOEIS learned to generate over more than 11 years is greater than one hundred thousands.

(Had I written this article today, I'd surely described the wonderful Sequence Machine which automatically generates conjectures relating sequences to each other, including really non-trivial ones, and the OEIS explorer which finds common terms in sequences.)

Regular contributors sometimes discuss ideas of sequence in a public mailing list called seqfan. The most qualified of them, such as the said Sean Irvine, are granted the editor status by Neil Sloane (accordingly, most of editors are not professional mathematicians). Some regular contributors are freaks whose perseverance is more like obsession; Neil Sloane bans the most odious of them.

Contributing to the OEIS is not the same as doing mathematical research, but ideally it means helping those who does mathematical research. Strathclyde University reader (now professor and associate dean) Sergey Kitaev, who made multiple contributions to the OEIS, and Rutgers University professor Doron Zeilberger, a member of the OEIS Foundation advisory board, say that the Encyclopedia is a well-known, respected, and valuable tool for mathematicians who work in the related areas (combinatorics, number theory) and for those who do recreational mathematics. Zeilberger did not miss the opportunity to note that recreational mathematics "is just as important and often more interesting" than mainstream mathematics; indeed, many of the OEIS materials may be not of great interest to mainstream mathematics, but they seem beautiful and interesting.

It is hard to say how often does the OEIS data help making non-trivial mathematical discoveries, but surely it happened many times. For example, Kitaev tells about sequence A022493 "Fishburn numbers <...>; also number of nonisomorphic interval orders on n unlabeled points":

The story began with me and two of my colleagues fooling around with a new notion of a permutation pattern (now known as bivincular patterns) and finding out using the OEIS that it is related to interval orders (which are the same as (2+2)-free posets). We've managed not only to explain combinatorially the connection, but also enumerate the interval orders via so called ascent sequences (we've introduced them as a tool, but now they gained substantial interest in the literature) thus solving a problem that was open around 40 years. Our work was very influential, and it resulted later in a family of nice generalizations of interval orders suggested by various authors. Essentially, none of this work would be possible without the OEIS, as how on Earth we would be able to guess that our pattern-avoiding permutations are connected to, say, interval orders?!

The On-Line Encyclopedia of Integer Sequences inspired other similar projects — "fingerprint databases". A typical example is the Encyclopedia of triangle centers by Clark Kimberling, Evansville University professor and a long-term OEIS contributor. That database begins with the incenter, the centroid, the circumcenter, and the orthocenter, and continues with the data on tens of thousands of points which could be called a triangle center in some sense. The influence of the OEIS is clearly revealed in this project in many aspects, from the organization of data to spartan design, and also in its huge size itself.

Future and challenges

Today (at the time of writing, i. e. 2020), as Russ Cox reports, OEIS has 300,000 visitors per month and 80,000 edits per year. When the presently used OEIS software was introduced in 2010, it allowed to distribute the huge work of processing all incoming edits, which had been earlier done solely by Neil Sloane, over several dozens of active editors. Jörg Arndt, a Technische Hochschule Nürnberg Georg Simon Ohm professor and one of the most active OEIS editors, notes: "Twenty years ago many entries were very terse, often the name just roughly indicating what the sequences is about and one or two articles cited. As of now many sequences include a self-contained definition and examples". Undoubtfully, this indicates success, as do thousands of references to the OEIS in literature. Neil Sloane was able to devote more time to mathematical work (researching sequences, writing papers), replying to mathematical and historical inquiries related to sequences using his big paper and electronic archives, more OEIS evangelism (giving talks and writing papers about new interesting sequences), and doing paperwork for the OEIS Foundation and strategic governance of the Encyclopedia.

What is the future of the OEIS in long-term perspective? When I asked Neil Sloane about it, he replied:

It is VERY IMPORTANT that it does not change too much. We have a very good system, and it is important that we preserve it. It has lasted for nearly 60 years. I hope that the database will still be around in a hundred years, without changing much. I have lived a long time, and many things have existed for a while, and then have been bought, or sold, or died, or disappeared. Or changed beyond recognition.

Also, Sloane believes that neither OEIS rules and guidelines nor its software need any changes. In future, the OEIS will be governed by the OEIS Foundation; the Foundations Board of Trustees for now has only dealed with financial questions, so apparently when Sloane ceases running the Encyclopedia, the Vice President of the Foundation will take over. Currently this position is held by Russ Cox. (In the end of June 2021 Russ Cox took over as the President of the Foundation. Sloane became the Chairman.) He also believes that the focus should be on the content of the Encyclopedia, and regarding the OEIS software he plans to continue improving the search and automate the index. We must note that Russ Cox indeed constantly improves the quality of the search over sequence data.

Does the OEIS really need no changes?

The growth rate of the OEIS is gradually increasing. On one hand, this indicates its success. On the other hand, it increases editors' burden. Neil Sloane rightly notes that waiting times for contributors are much shorter than in top mathematical journals. However, he regularly says in his talks that the OEIS is swamped with submissions and needs more editors. The draft limit for new contributors was recently lowered from 7 to 3 for a reason. Indeed, although edits from regular contributors and editors are often published quickly, many drafts wait in the queue for a long time. There must be some reasons why editors pass by these drafts all the time and the most effective decision-making mechanism is Sloane ex machina.

These reasons are diverse and intertwined. The OEIS wiki engine lacks many features that are present in MediaWiki and to which any Wikipedia editor is used to, such as a watchlist or search over talk pages (pink-box comments). Within a particular draft, all comments and outdated versions are shown altogether; in nontrivial cases they snowball, demotivating editors from digging into long discussions and also preventing them from expressing their own opinions to avoid complicating the work for others (I think this especially affects Associate Editors).

Ever since the Encyclopedia wasn't On-Line, a reasonable rule has been set: each comment and formula in sequence entries should contain a dated signature of its contributor. The wiki engine with the convenient edit history browsing would normally make the signatures obsolete, but the rules still require to add them. Also, the OEIS rules say that editors could "edit mercilessly" any contributed text. As a result, much effort is spent on arguments around processing, say, signatures under comments written by one contributor and changed by another one. Also, the chronological order is used in the comments, which prevents their organization and topical grouping.

Furthermore, different editors have different tastes and slightly different vision of the concept of a sequence being interesting and appropriate. Also, Neil Sloane says that setting up "laws that handle every possible situation" is not in the spirit of the OEIS. That is true, but the lack of written universally accepted principles leads to complications and delays in processing sequences that are not very interesting. The stream of borderline-interesting sequences is significant, they litter the draft queue, and when they make it into the OEIS they litter the search results, becoming the problem not for editors but for readers.

Few unequivocally set rules are related mostly to style. They are not always written in the OEIS Wiki, and even when they are, there are still no attention-grabbing links to them from the OEIS user interface, which generates a large stream of identical troubles with different contributors' drafts and, accordingly, identical comments from editors: "please add a dot at the end of the title" etc. Since some of these rules and restrictions are very simple and formal (e. g.: the b-file and the data in the main sequence entry must be consistent with each other), they could be implemented by purely technological means, but the OEIS wiki engine does not allow that, and Neil Sloane usually rejects technological approaches for performing routine tasks.

Georg Fischer, who had worked in IT for 45 years and has recently joined the work on jOEIS, has developed tools for automated checks for the OEIS for obvious errors such as inconsistent b-files. He rightly thinks that these tools should be integrated into the OEIS software. He also notes that the OEIS user interface needs renovation. And if one is going to make a list of changes that many editors (including me) consider critically needed, it would be so long that the most relevant tasks are meta-improvements: Georg Fischer named extending the team of server admins and programmers as the most urgent task, Jörg Arndt named launching a bug tracking system for the OEIS software. (The OEIS software issue tracker now exists on Github. It has been created by Cox. However, the issues have received zero visible reaction so far.) Possible security problems deserve special mention: if vulnerabilities are discovered in the OEIS software, which has never been studied by information security experts, and malicious agents exploit them, then the Encyclopedia end users could be hurt.

Currently the development of the OEIS software is done by its author, Russ Cox. He has written the source code for this system using Go programming language. It was a natural choice since Russ himself develops this language; he still leads the development of that project at Google (which probably doesn't help devoting much free time to a hobby such as the OEIS). But then, in 2010, that language was new, there were little programmers who could use it. That raised the barrier for new developers to working on the OEIS software. Now Go is more widespread, so more developers could be brought; the OEIS Foundation has no money to pay them but everybody who works on the OEIS is a volunteer anyway. Russ Cox says that "we took some steps [in 2019] toward making the software something that more people could collaborate on, but we're still not there yet". So Russ Cox doesn't view recruiting new developers as a high priority, neither does Neil Sloane. Charles Greathouse says that there is a small group of people who want and can make changes in the OEIS software but don't do that for various reasons (Greathouse himself says he doesn't have enough time). Therefore, changes that many editors consider necessary does not seem to be happening soon. As of summer 2022, the OEIS software developers team still consists of Russ only.

And there is a meta problem: the OEIS community just has no practice of discussing such issues and making any decisions as a result of such discussions. Unlike Wikipedia, the OIES has no message boards; editors' mailing lists have little activity, and anyway only specific issues are discussed there, while attempts to discuss general issues result in nothing; the OEIS Foundation Board of Trustees only deals with financial matters; the Foundation Advisory Board does not function. As a result, only Neil Sloane himself can make strategic decisions, and he has little time to discuss such matters and a rather conservative own position.

Neil Sloane has created a wonderful Encyclopedia which unites fans of mathematics from all around the world and became a valuable tool for professional mathematicians. He always has kind words for contributors and editors. Russ Cox made possible the development of the Encyclopedia in its current form and continues to improve it. (There are other people involved in the creation and functioning of the OEIS; unfortunately, we have no space here to describe their roles.) Contributors continue extending the OEIS, while editors maintain the quality of its content. I hope that the On-Line Encyclopedia of Integer Sequences will live and develop happily ever after.