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Welcome to The On-Line Encyclopedia of Integer Sequences® (OEIS®) Wiki

Some Famous Sequences

Click on any of the following to see examples of famous sequences in the On-Line Encyclopedia of Integer Sequences (the OEIS), then hit "Back" in your browser to return here:

For some other fascinating sequences see Pictures from the OEIS: The (Free) OEIS Store

General Information About OEIS

  • Most people use the OEIS to get information about a particular number sequence. If you are a new visitor, then you might ask the database if it can recognize your favorite sequence, if you have one. To do this, go to the main look-up page, enter the sequence, and click Search. You could also look for your sequence in the Index.
  • If your favorite sequence isn't in the database, and if it is interesting, please submit it using the web page for Contributing a new sequence or comment. Of course the sequence should be well-defined, of general interest and ideally it should be infinite. Short sequences such as phone numbers are not appropriate. Before submitting a new sequence, you must first register.
  • If you have stumped the database, you can try Superseeker, which tries really hard to identify a sequence. Send an empty email message to [email protected] to get instructions.
  • You can browse the database, using the WebCam. This can be set to look at the most interesting sequences, recent additions, or sequences needing more terms. It can be quite addictive!
  • It is also interesting to browse the Index to the OEIS to see the variety of topics that are covered. In a way the OEIS can be regarded as an index to all of science. It is like a dictionary or fingerprint file for number sequences.

Introductory chapters from the 1973 and 1995 books; Supplement 3 to 1973 book

The introductory chapters from N. J. A. Sloane's Handbook of Integer Sequences (1973) (click here) and N. J. A. Sloane and S. Plouffe's Encyclopedia of Integer Sequences (1995) (click here) contain much useful information about analyzing an unknown sequence and many other topics.

The third Supplement to the 1973 Handbook (which supersedes the first two supplements) is available here: Media:Supp3.txt

Description of OEIS entries (or, What is the Next Term?)

What comes next after 1, 2, 4, 9, 20, 48, 115, 286, 719, ... (for example)? This is the place to find out! (Answer: click here.)

  • The main table in the OEIS is a collection of over a quarter-million number sequences. The entry for each sequence gives some or all of:
    • the beginning of the sequence
    • its name or description
    • a graph of the sequence
    • additional comments
    • the offset (index of first term)
    • references or links
    • formulas
    • computer programs
    • cross-references to other sequences
    • the sequence converted to music
    • the name of the person who submitted it
    • the history of the OEIS entry
  • For further information about the format of replies received from the database, click here. See also the hints file for further useful information.
  • The sequence pages have little buttons at the top marked "list", "graph", "refs", "listen", "history", "text", "internal format", and sometimes "table" and "edit".
    • "list" produces a numbered list of the terms, plus a bracketed list suitable for importing into other programs such as Maple, Mathemtica, PARI, etc.
    • "refs" shows all the sequences which reference this one.
    • "table": If the sequence is formed by reading a triangle across rows (or by reading a table by antidiagonals), this button produces three different two-dimensional views of the sequence. For an example, see Pascal's triangle, A007318.
    • "graph" produces two plots of the sequence. The first is a pin plot of the first 200 terms (less if fewer terms are available), the second is a linear or log scatter-plot of all available terms, using terms from the b-file if there is one. Some noteworthy plots are the Fibonacci numbers A000045, the partition numbers A000041, the Euler phi-function A000010, etc.
    • "listen" produces a midi file so that you can listen to the sequence. The first time you use it you will probably have to tell your browser to allow popups from the OEIS web site. Try listening to Recaman's sequence A005132, turn the volume up to 127 and set the instrument to #103 !
    • "history" shows the history of changes to this entry.
    • "edit" (you will see this only if you are a registered user) enables you to propose changes to the sequence.
    • "text" shows the various sections of the entry for this sequence with each line prefixed by a symbol that indicates its function.
    • "Internal format" shows the various sections of the entry for this sequence in the internal format.

OEIS: Brief History

The sequence database was begun by Neil J. A. Sloane (henceforth, "NJAS") in early 1964 when he was a graduate student at Cornell University in Ithaca, NY. He had encountered a sequence of numbers while working on his dissertation, namely 1, 8, 78, 944, ... (now entry A000435 in the OEIS), and was looking for a formula for the n-th term, in order to determine the rate of growth of the terms.

He noticed that although several books in the Cornell library contained sequences somewhat similar to this, this particular sequence was not mentioned. In order to keep track of the sequences in these books, NJAS started recording them on file cards, which he sorted into lexicographic order.

Here is a scan of the page in NJAS's thesis notebook with the very first collection of sequences. (The sequences mentioned are A000027, A000217, A000292, A000332, A000389, A000579, A000110, A007318, A000058, A000215, A000289, A000324, A234953 (= A001854(n)/n), A000435, A000169, A000142, A000272, A000312, A000111.) This is the acorn from which the OEIS grew. The date is January or February, 1964.

The sequences were transferred to punched cards in 1967, and were made into a book in 1973 ("A Handbook of Integer Sequences", by NJAS, Academic Press, NY). This book contained 2372 sequences.

[Incidentally, today these cards are sometimes referred to as "punch cards" (sic). This is wrong, they were always called "punched cards". Anyone who says "punch cards" is showing they know nothing about the subject. Saying "punch cards" is like saying "hike boots" or "walk stick" or "chew gum".]

NJAS joined AT&T Bell Laboratories in 1969. Following the publication of the book, a large amount of correspondence ensued, with suggestions for further sequences and updates to the existing entries. Many people remarked how useful they found the book, and how surprising it was that no one had published such a collection before.

By the early 1990's over a cubic meter of correspondence had accumulated. A Canadian mathematician, Simon Plouffe, offered to help in preparing a revised edition of the book, and in 1995 "The Encyclopedia of Integer Sequences", by NJAS and Simon Plouffe, was published by Academic Press, San Diego. It contained 5487 sequences, occupying 587 pages. (Incidentally, Simon Plouffe is now one of the Trustees of The OEIS Foundation Inc..)

Again, once the book appeared, many further sequences and updates were submitted from people all over the world. NJAS waited a year, until the size of the collection had doubled, to 10000 entries, and then in 1996 he launched The On-Line Encyclopedia of Integer Sequences® (OEIS®) on the Internet. From 1996 until October 26, 2009, this was part of NJAS's home page on the AT&T Labs website.

During this period, from 1996 to 2009, the database grew by at least 10000 entries per year (18000 new sequences were added in 2009 alone). If it were to be published in book form today, the OEIS would require over 750 volumes, each the size of the 1995 book.

Starting in 2002, NJAS added a group of associate editors to help process submissions. However, because they did not have access to the computer where the database was maintained, almost all the work of updating had to be done single-handedly by NJAS. This involves processing 100 or 200 emails every day, and was getting to be beyond what one person can handle.

In 2009, therefore, it was decided to make a drastic change. NJAS set up a non-profit foundation, The OEIS Foundation Inc., whose purpose is to own, maintain and raise funds to support The On-Line Encyclopedia of Integer Sequences® (OEIS®). On October 26, 2009, NJAS transferred the intellectual property of The On-Line Encyclopedia of Integer Sequences to the Foundation and work was begun on moving the database from NJAS's home page at AT&T to a commercial hosting service.

Here, however, we ran into a very serious problem. In the summer of 2009, when we tried to get the OEIS working as a wiki, we discovered that the Mediawiki software was not capable of handling the kind of queries that arise in looking up sequences. This was a disaster.

It took us over a year to resolve this problem. In the end, Russ Cox completely rewrote all the programs needed to maintain the database and answer queries - a huge task! NJAS's colleague David Applegate has also been of enormous help in getting the new system working. As a result of their work, the new OEIS was finally launched on November 11, 2010. It is now possible for anyone in the world to propose a new sequence or an update to an existing sequence. To do this, users must first register. A group of about 130 editors has been formed, whose job it is to review submissions before they become a permanent part of the OEIS.

So, after nearly two years of struggle, the OEIS was finally able to operate without NJAS having to approve every change. After 46 years of running the database, this came as a great relief to him.

OEIS: The Movie

To celebrate the launching of The OEIS Foundation Inc, Tony Noe made an 8.5-minute movie showing graphs of the first 1000 terms of 1000 sequences, with soundtrack from Recaman's sequence A005132.

There are four ways to view the movie:

  • On YouTube (you can find it by searching for "OEIS" and "Movie").
  • By downloading a 5 MB QuickTime movie that is viewable with QuickTime Player 7 and some browsers.
  • By downloading a 27 MB movie that uses the H264 codec and AAC sound. This movie is viewable on recent versions of Windows Media Player and most up-to-date browsers.
  • By going to Tony Noe's website for a frame-by-frame display, with links to the definitions of the sequences.

Arrangement of the Sequences in Database

  • Sequences in the OEIS are arranged in lexicographic order, indexed by the position of the first term that is greater than 1 in absolute value. Sequences that contain only 0's, 1's and -1's are in lexicographic order by absolute value at the beginning of the table.
  • Thus there is an essentially unique place to look in order to see if a sequence is already in the table. (If it isn't, submit it and it will probably be added if it is sufficiently interesting - see contributing a new sequence or comment.)
  • Each entry in the OEIS has a link called Sequence in context, which shows the three sequences immediately before and after it in the lexicographic order. (If you don't see it, click on the A-number.)
  • There is also a link called Adjacent sequences, which shows the three entries whose A-numbers are immediately before and after the current sequence.
  • These two links can be very useful when you are looking for a sequence in the OEIS but you are not sure of some of the terms.

Format Used in Replies From the Database

For information about the format of replies received from the database, click here. See also the hints file for further useful information.

Index

  • There is an Index to the most important sequences.
  • The main look-up page will also allow you to search for a word (or do much more complicated searches) in the database.

Sequences Which Agree For a Long Time

  • People are always asking about this, so there is a section about them in the Index.

Recent Additions

  • Recent additions to the OEIS can be seen by clicking the Recent Additions link.
  • You can also browse the recent additions using the WebCam.

Compressed Versions

  • There is a gzipped file containing just the sequences and their A-numbers (a few tens of megabytes)
  • There is also a gzipped file containing just the names of the sequences and their A-numbers (a few megabytes)

These two files are updated daily.

Contributing a New Sequence, Comment or More Terms

OEIS Search Bar

To add an OEIS search bar to your browser, see the instructions here.

Email Addresses, Getting in Touch With Authors

The old OEIS on NJAS's homepage gave email addresses (in disguised form) for all contributors. This facility is essential in a scientific database, in order that questions involving definitions, possible errors, etc., can be discussed. In the current OEIS Wiki, however, email addresses are not made public and a different mechanism is used for contacting contributors.

First, find the author in the list of Registered Users and go to the author's User Page. (Make sure you are logged in to the OEIS Wiki - this is different from logging in to the OEIS database). There you will see a button in the left panel saying "Email this user".

Ombudsman

Hilarie Orman has volunteered to serve as an ombudsman to help resolve disputes with contributors. She can be reached through her user page on this wiki.

Sequences in Classic Books

This page has moved. Please go to Sequences From Classic Books.

Citations

Referencing the OEIS

If you have found the OEIS useful and wish to reference it, the usual citation is

OEIS Foundation Inc. (2024), The On-Line Encyclopedia of Integer Sequences, Published electronically at https://oeis.org.

URLs

Referencing a Particular Sequence

  • If you are writing a paper and wish to refer the Catalan numbers, say (sequence A000108), but don't want to digress to describe them, simply add a reference or link that points directly to that sequence in the OEIS.
  • A text reference might say:

The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org, 2010, Sequence A000108

or, if it is clear who "discovered" the sequence, something like

J. H. Conway, Sequence A007970 in The On-Line Encyclopedia of Integer Sequences (2010), published electronically at https://oeis.org.

  • In an HTML file one might say something like this: ... where the C(n) are the Catalan numbers (<a href="https://oeis.org/A000108">Sequence A000108</a> in [OEIS]).

Acknowledgments

  • A very large number of people have contributed to the database, and it would be impossible to thank them individually.
  • Special thanks to Antti Karttunen, who wrote the program that displays sequences based on arrays (those with keyword "tabl") in three different two-dimensional formats. To see this, look at some of the following sequences, and click on the keyword "tabl":

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Links to Other Sites

OEIS Mentioned in WolframAlpha Timeline

Awards and press clippings

See the separate Awards and press clippings page for the full list back to the 1970s.

  • The TV series Mr Robot mentioned the OEIS in Season 2, Episode 11 or 12 (there are conflicting reports), September 2016, about 37 minutes in. They are deciphering a cryptic message with the aid of the OEIS. The scene lasts for two to three minutes (which is a lot of screen time).
  • The Dutch magazine Pythagoras has an ongoing series of articles about number sequences, many of which mention the OEIS. Four parts have appeared so far: Een Lexicon vol Getallen ["A Dictionary of Numbers"] (Sept. 2015), Getallenplantjes ["Number plants" (?)] (Oct. 2015), Driehoeksgetallen ["Triangular numbers"] (Nov. 2015), Een bizarre rij [A bizarre sequence] (Dec. 2015).
  • Margaret Wertheim, The Fax Numbers of the Beast, and Other Mathematical Sports: An Interview with Neil Sloane, Cabinet Magazine, Issue 57, Spring 2015, pages 48-54.
  • Siobhan Roberts, How to Build a Search Engine for Mathematics: The surprising power of Neil Sloane's Encyclopedia of Integer Sequences, Nautilus Magazine, Issue 29, Chapter 4, Oct 22 2015.
  • Eric Egge in Defying God: The Stanley-Wilf Conjecture, Stanley-Wilf Limits, and a Two-Generation Explosion of Combinatorics, pp. 65-82 of A Century of Advancing Mathematic, ed. S. F. Kennedy et al., MAA Press 2015, says: "When we enter our terms 1, 2, 6; 22, 91, 408, 1938, 9614 into the OEIS search box, we are rewarded with a description of the sequence .... In hindsight it’s amusing that Sloane called his [1973] book “A Handbook,” as though there might be competitors. There are none, and the OEIS is a required stop for anyone who encounters an integer sequence they don’t recognize. It’s no exaggeration to observe that in certain parts of combinatorics, the OEIS alone has increased the rate of new discoveries by an order of magnitude."
  • Featured by Slashdot on August 7 2015
  • Erica Klarreich, The Connoisseur of Number Sequences, Quanta Magazine, August 6 2015 (Interview with Neil Sloane about the OEIS)
  • Hacker News discussion of OEIS, July 21 2015
  • Article by Vijayakumar Ambat in Malayalam (a regional language of India) in the newspaper Malayala Manorama - Padhippura, 12 June 2015, that mentions the OEIS.
  • Video interview about the OEIS by Marc Chamberland made at the Joint Math Meetings in San Antoio, Texas, Jan. 2015.
  • Alex Bellos, Neil Sloane: the man who loved only integer sequences, Alexs-adventures-in-numberland blog, The Guardian, Oct 07 2014.

Copyright Notice

This database and its associated files are copyright 2024 by The OEIS Foundation Inc..

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