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Welcome to The On-Line Encyclopedia of Integer Sequences® (OEIS®) Wiki
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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:
- Recamán's sequence, A005132
- The Busy Beaver problem, A060843
- The Catalan numbers, A000108
- The prime numbers, A000040
- The Mersenne primes, A000043 and A000668
- The Fibonacci numbers, A000045
For some other fascinating sequences see the OEIS Poster.
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 Submit. You could also look for your sequence in the Index.
- You might also look at the demonstration pages to see more examples of how to use the OEIS.
- 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 superseeker@oeis.org 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.
- Also worth visiting are the pages dealing with Puzzle sequences, Classic sequences and Hot sequences.
- You might also like to see a list of papers that have acknowledged help from the database and some comments from readers.
Introductory chapters from the 1973 and 1995 books
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.
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 about 200000 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", "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.
- "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.
Since Nov 11 2010 the OEIS has been running as a moderated wiki on its own web site, http://oeis.org. The conversion to a wiki — a monumental task — was carried out by Russ Cox. David Applegate is the OEIS webmaster. See the OEIS Foundation http://oeisf.org for more information about the history of the OEIS and its conversion to a wiki. Users of the OEIS are urged to donate to the OEIS Foundation!
OEIS: The Movie
To celebrate the launching of The OEIS Foundation Inc, Tony Noe has 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.
(Incidentally, you can convert the movie to just about any other format at http://www.media-convert.com, without downloading any software).
Arrangement of 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 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 (about 9 megs)
- There is also a gzipped file containing just the names of the sequences and their A-numbers (about 3 megs)
Contributing a New Sequence, Comment or More Terms
- For contributing a new sequence, comment or more terms for an existing sequence, see this page.
- Want to help?
- See the web page Sequences that need more terms
- Or use the WebCam to browse the sequences that need more terms
- or use the main look-up page to search for the keyword more
- See also the page future projects (this page needs updating)
OEIS Search Bar
To add an OEIS search bar to your browser, see the instructions here.
Email Addresses, Getting in Touch With Authors
The Classic OEIS provides email addresses (in disguised form) for all contributors. This is essential in a scientific database, in order that questions involving definitions, possible errors, etc., can be discussed. In the OEIS Wiki, however, email addresses will not be made public. A different mechanism will be used for getting in touch with authors.
First, find the author in the list of Registered Users and go to the author's User Page. Click the Discussion Tab to go to the User Talk page. Then click on the Edit/+ Tab to enter a message. Sign the message by typing ~~~~ (four tildes) when you are done. Then be sure to click both Preview and Save Page. This should trigger an email message to the author.
There are several web pages that provide advice on editing Talk Pages. See for example Wikipedia's Help:Using Talk Pages.
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
These files contain concordances to the sequences mentioned in certain classic books.
- L. Comtet's Advanced Combinatorics
- P. Flajolet and R. Sedgewick's Analytic Combinatorics (Does not yet exist)
- R. L. Graham, D. E. Knuth and O. Patashnik's Concrete Mathematics
- F. Harary and E. M. Palmer's Graphical Enumeration
- R. P. Stanley's Enumerative Combinatorics.
Citations
- Papers Citing the Encyclopedia of Integer Sequences. Shows some of the ways that people have used the database.
Referencing the OEIS
If the database helped your work and you wish to reference it, the usual citation is something like this:
The On-Line Encyclopedia of Integer Sequences, published electronically at http://oeis.org, 2010.
URLs
- The URL for the main lookup page is http://oeis.org
- The URL for the present page is http://oeis.org/wiki/Welcome
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.
- The URL for sequence A000108 (for example) is http://oeis.org/A000108.
- A text reference might say:
The On-Line Encyclopedia of Integer Sequences, published electronically at http://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 http://oeis.org.
- In an HTML file one might say something like this: ... where the C(n) are the Catalan numbers (<a href="http://oeis.org/A000108">Sequence A000108</a> in [OEIS]).
Policy on Searching the Database
- Just as it is OK for a browser (such as Firefox) to access the OEIS, so it is also OK for a computer algebra program such as SAGE or Haskell to have an interface with the OEIS, provided of course that this does not put too much of a burden on the server here. On the other hand, it would definitely not be OK to distribute a copy of the OEIS with such a program.
- See The OEIS End-User License Agreement for further information.
Acknowledgments
- A very large number of people have contributed to the database, and it would be impossible to thank them individually. Their names can be seen throughout the nearly 200000 entries.
- 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":
- At the end of 2005, Alex Healy and Russ Cox (rsc(AT)swtch.com) made a huge contribution to OEIS by greatly speeding up the search process in the Classic OEIS. The first versions of the new programs were written by Alex Healy and the final versions by Russ Cox. David Applegate helped install them. The new searches were much faster than the old ones and can handle much more complicated queries. See the hints file for details.
- During 2009-2010, David Applegate has been working on the enormous tasks of (1) moving the OEIS from Neil Sloane's home page at AT&T to its new home at oeis.org/classic, and (2) converting it to a wiki format at oeis.org/wiki. The second step has turned out to be unexpectedly difficult, because we discovered that the Wikimedia software is unsuitable for performing the kind of searches needed for the OEIS. Russ Cox has also been helping solve this problem. It is hoped that this work will be completed in the summer of 2010.
Related OEIS Pages
Some of these are links to the Classic OEIS. They will be gradually be replaced by links to the OEIS Wiki. Also several of these pages still need to be converted from html to wikimedia.
- Main look-up page
- Lookup hints
- Welcome page (this page)
- Index to OEIS
- Classic OEIS
- OEIS in other languages (use the Classic OEIS)
- The OEIS Foundation Inc.
- Register (an account is necessary to contribute to the Wiki).
- Registered Users
- Contributing a new sequence, comment or more terms for an existing sequence
- Editorial Board
- Format (1), Format (2)
- Transformations of sequences
- Maple or Mathematica (see EISFormat.m) scripts to format sequences
- OEIS FAQ, Wiki FAQ
- OEIS 100K E-Party
- OEIS MOVIE
- Demonstration pages
- OEIS WebCam
- Sequences that need more terms
- Future projects (this page needs updating)
- Index to Fractions
- Find Friends
- Music (this page needs updating)
- Plot one sequence against another
- Puzzle sequences, Classic sequences, Hot sequences (these pages need to be updated)
- More Pages (Wiki version), More Pages (Classic version)
Links to Other Sites
- b-file Maple programs written by R. J. Mathar
- Caldwell's Prime Pages
- Combinatorial Object Server
- Davalan's Jeux et Mathématiques
- De Geest's World of Numbers
- Encyclopedia of Combinatorial Structures
- Finch's Mathematical Constants
- Geometry Junkyard
- Journal of Integer Sequences
- MathSciNet
- The Nth Prime Page
- Plouffe's Inverter (see also the Inverse Symbolic Calculator)
- SeqFan Mailing List
- Primo
- Neil Sloane's home page
- Eric Weisstein's MathWorld
OEIS Mentioned in WolframAlpha Timeline
- The OEIS appears (under 1973) in WolframAlpha's Timeline of Systematic Data and the Development of Computable Knowledge, which extends from cave paintings to the present day.
Awards and press clippings
See Awards and press clippings for the full list back to the 1970s.
- 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.
Copyright Notice
This database and its associated files are copyright 2015 by The OEIS Foundation Inc..