The On-Line Encyclopedia of Integer Sequences

OEIS, the On-Line Encyclopedia of Integer Sequences, is probably the authoritative source for information on integer sequences, which arise primarily from combinatorics, number theory, and recreational mathematics. However, the often deep connections between mathematical patterns and those in the real world mean that numerous sequences can be identified in terms of physical or chemical phenomena, and most branches of mathematics are represented in some fashion.

It is easily the largest such collection available, anywhere. As of 2004, there are over 100,000 entries in the database, which can be queried by web-search (at http://www.research.att.com/~njas/sequences) or via an email service (details at http://www.research.att.com/~njas/sequences/ol.html). As the URLs suggest, the project is hosted by AT&T research; however, it is largely the product of 40 years of collecting by mathematician Neil Sloane. Two books of his collection have been released, in 1973 and 1995; but the subsequent influx of sequences has rendered the project unmanageable as a printed work and since 1996 the complete database has existed only as a (free) internet resource, maintained primarily by Sloane but supported in recent years by volunteers and a board of editors. The site grows at a rate of about 10,000 sequences a year.

An entry for a sequence offers far more than just a name and some terms. A stated aim of the OEIS is to benefit active researchers in the sciences and mathematics, both by simplifying work through supplying general formulae or lists of terms and by highlighting previous work on a given sequence through journal articles, books, websites and related sequences. Each sequence has an ID number and URL on the OEIS for ease of reference and citation.

The standard search process is to simply drop in the first half dozen or so terms into the web search page. However, advanced searching on the text of the entries is possible (to find work by a given author or in a certain field); and there is a super-seeker that tries particularly hard to match sequences by applying various transformations to the supplied terms, although due to the additional computational overhead it is requested that its usage be kept to a minimum.

In addition to being a crucial academic resource, the database is a goldmine for curious mathematical diversions. It's possible to browse for random entries, drawn either from the entire database or the "best" highlights. A puzzles page challenges you to determine the rule behind a given collection of terms. The "hot" page picks out some of the best and worst recent additions- which can also be explored with the 'webcam', which serves up recent contributions every few seconds until you spot one you like... as the frontpage observes, all this can become strangely addictive...

Reference

• The OEIS itself, frontpage and FAQ.
  http://www.research.att.com/~njas/sequences/
• Notices of the AMS, Vol.50 No.8: article by Neil Sloane on the project.
  http://www.ams.org/notices/200308/comm-sloane.pdf
• The 1995 print version, Encyclopedia of Integer Sequences has ISBN 0125586302 and can be found on, amongst others, Amazon.

Holy mole! The last writeup on this node is almost 12 years old! A lot has happened since.

The On-Line Encyclopedia of Integer Sequences is an online database of (have you guessed it?) sequences of Integers (the set of numbers commonly denoted by ℤ) founded by Neil Sloane.¹

The encyclopedia began in 1964. Sloane was working on his dissertation, when he came across a certain sequence (1, 8, 78, 944…)² and was looking for a formula to derive the n-th number in the sequence. His research at the Cornell University library led him to similar sequences, though not the exact one he was looking for. Since smartphones weren’t around then, Sloane recorded these sequences in file cards sorted lexicographically.

The file cards were eventually transferred to punched cards and, in 1967, to a book with 2,372 sequences.³

In 1969 Sloane started working at AT&T Bell Laboratories and over the course of the next 21 years accumulated «a cubic meter of correspondence»⁴ with praise, further suggestions and updates to the sequences of the book.

In 1995, with the help of Simon Plouffe, the Encyclopedia of Integer Sequences was published⁵ with 5,487 sequences printed on 587 pages. Amazing! But it wouldn’t stop there.

Once more, Sloane received numerous correspondence from people around the world. This time the collection nearly doubled in size in a year, so in 1996 the magnificent On-Line Encyclopedia of Integer Sequences was born, hosted at the AT&T servers.

However, the entries just kept coming. Between 1996 and 2009 the database grew by more than 10,000 sequences per year. There was also the problem that only Sloane had access to the database and so it was increasingly difficult to maintain the database. In 2002 a group of associate editors started helping but it wasn’t enough. Another big leap had to be done.

On October 27, 2009 Sloane transferred all of his intellectual property and hosting to the OEIS Foundation Inc.⁶ The original url⁷ now resolves to the error code 503 Service Unavailable. Now, the OEIS can be reached at https://oeis.org/

Last modified May 8 20:41 EDT 2017. Contains 286111 sequences.

More than sequences

The OEIS hosts more than just a database of sequences. A complete entry at the OEIS has:

Sequence of integers

y'know, the actual sequence of numbers

Offset

Value and index of the first members of the sequence

Comments

An English version of the sequence, in case the title is unreadable. Sometimes these are also unintelligible without knowledge of advanced mathematics, but at least are human-readable. Related sequences also go here.

References

Papers or books where the sequence is described and/or used

Links

Out-links to the big blue wet internet

Formula

When there is one, a mathematical way of generating the numbers

Example

Mathematica

Code for displaying the sequence in Wolfram’s Mathematica

Prog

Code for generating the sequence in some programming language

Keywords

One or more specific keywords describing properties of the sequence.

Author

Extensions

More credits for people who have extended, corrected or edited by someone else.

More! More!

Also at the OEIS:

• Puzzle Sequences!
• Generate musical sequences from numerical sequences!
• Plot pairs of sequences!
• The WebCam, randomly jumping between sequences!
• The (Free) OEIS Store!
• Wanna help?

Even More!

• Interesting sequences
  • Busy Beaver Sequence
  • Recamán’s sequence
  • Catalan numbers
  • Mersenne primes
• Interesting people
  • Neil Sloane
  • Paul Erdös
  • John Conway
  • Donald Knuth

1. From Wikipedia: Neil James Alexander Sloane (born October 10, 1939) is a British-American mathematician. His major contributions are in the fields of combinatorics, error-correcting codes, and sphere packing. ↩

2. Now formalized in the OEIS as A000435: “Normalized total height of all nodes in all rooted trees with n labeled nodes. http://oeis.org/A000435↩

3. N. J. A. (1973). A handbook of integer sequences. New York: Academic Press. ISBN 978-0-12-648550-9.↩

4. https://oeis.org/wiki/Welcome↩

5. Sloane, N. J. A. (1995). The encyclopedia of integer sequences. San Diego: Academic Press. ISBN 0-12-558630-2.↩

6. http://oeisf.org/index.html#IPXFER↩

7. http://www.research.att.com/~njas/sequences/ol.html↩