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

• Other pages: The OEIS Foundation.   Use database.   Editors   Run demo.   Sequence webcam.   Index.   FAQ.   Format (external),   (internal).   Versions in other languages.   Recent additions (possibly a large file).   Index to fractions.   Works citing OEIS.   OEIS: The 100K E-party.
• Contents of this page:   New users.   Description of the database.   OEIS: The movie.   OEIS: Brief history.   Arrangement of sequences in database.   Gzipped version.   Gzipped names.   Contributing new sequence or comment; helping.   OEIS search bar.   Sequences in classic books.   Referencing OEIS.   URLs.   Referencing a particular sequence.   URL for searching the database.   Policy on searching the database.   Policy on email addresses.   Copyright Notice.   Acknowledgments.   Links.   Awards, etc.

• New Users.
  • Let's begin at once with an example of a sequence of great importance:

    A060843 Busy Beaver problem: maximal number of steps that an n-state Turing machine can make on an initially blank tape before eventually halting. +30 5

    1, 6, 21, 107 (list; graph; listen)

    OFFSET 1,2 COMMENT "In 1965 [Tibor] Rado, together with Shen Lin, proved that BB(3) is 21. ... Next, in 1983, Allan Brady proved that BB(4) is 107. ... Then, in 1989, Heiner Marxen and Juergen Buntrock discovered that BB(5) is at least 47,176,870. ... As for BB(6), Marxen and Buntrock set another record in 1997 by proving that it is at least 8,690,333,381,690,951." Aaronson. The function Sigma(n) (A028444) denotes the maximal number of tape marks which a Turing Machine with n internal states and a two-way infinite tape can write on an initially empty tape and then halt. The function S(n) (the present sequence) denotes the maximal number of steps (shifts) which such a machine can make (it needs not produce many tape marks). Given that 5-state machines can compute Collatz-like congruential functions (see references), it may be very hard to find the next term. The sequence grows faster than any computable function of n, and so is non-computable. REFERENCES Brady, A. H., The busy beaver game and the meaning of life, in Herken, R. (Ed) The Universal Turing Machine, pp. 259-277, Oxford Univ Press 1988. Brady, A. H. The determination of Rado's noncomputable function Sigma(k) for four-state Turing machines, Math. Comp. 40 #62 (1983) 647-665. Machlin, R. (nee Kopp), and Stout, Q, The Complex Behavior of Simple Machines, Physica D 42 (1990) 85-98 Michel, Pascal, Busy beaver competition and Collatz-like problems, Arch. Math. Logic (1993) 32:351-367. R. M. Robinson, Minsky's small universal Turing machine, Int'l Jnl. Math, 2 #5 (1991) 551-562. Yu. V. Rogozhin, Seven universal Turing machines (Russian), abstract, Fifth All-Union Conference on Math. Logic, Akad. Nauk. SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1979, p. 127. Yu. V. Rogozhin, Seven universal Turing machines (Russian), Systems and Theoretical Programming, Mat. Issled. no. 69, Akademiya Nauk Moldavskoi SSSR, Kishinev, 1982, pp. 76-90. Claude E. Shannon, A universal Turing machine with two internal states, Automata Studies, Ann. of Math. Stud. 34 (1956) 157-165. LINKS Scott Aaronson, Who Can Name the Bigger Number? Bill Dubuque, Re: Halting is weak A. Gravell and U. Ultes-Nitsche, BB(n) Grows Faster Than Any Computable Function H. Marxen, Busy Beaver Problem M. Somos, Busy Beaver Turing Machine M. Somos, Busy Beaver Q. F. Stout, The Complex Behavior of Simple Machines E. W. Weisstein, Link to a section of The World of Mathematics. E. W. Weisstein, Busy Beaver Index entries for sequences related to Busy Beaver problem CROSSREFS Cf. A028444. Sequence in context: A012662 A012418 A083558 this_sequence A026650 A009253 A012840 Adjacent sequences: A060840 A060841 A060842 this_sequence A060844 A060845 A060846 KEYWORD hard,nice,nonn,bref AUTHOR Jud McCranie (JudMcCranie(AT)ugaalum.uga.edu) and njas, May 02 2001 EXTENSIONS The next two terms are at least 47176870 and 3*10^1730. Additional references from Bill Dubuque (wgd(AT)martigny.ai.mit.edu)

    page 1

  • Most people use this web site 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. (Of course, the number sequence should be well-defined, of general interest and ideally it should be infinite. Short sequences such as phone numbers are not appropriate.)
  • If your 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.
  • If you have stumped the database, you can try Superseeker, which tries really hard to identify a sequence.
  • 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 interesting to scan the Index to the database to see the variety of topics that are covered. In a way this database 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 can run the demonstration pages to see more examples of how to use The On-Line Encyclopedia of Integer Sequences.
  • The Search pages have little buttons called "list", "graph", "listen" and sometimes "table".
    • "List" produces a numbered list of the terms, plus a bracketed list suitable for importing into other programs.
    • "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 scatterplot 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. The plotting program was written by Deborah Swayne using the R language.
    • "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 web site. Try listening to Recaman's sequence A005132, turn the volume up to 127 and set the instrument to #103 !
    • "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.
  • Finally, you might like to see a list of papers that have acknowledged help from the database and some comments from readers.

• Description of the Database (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!
  The main table is a collection of number sequences arranged in lexicographic order. The entry for each sequence gives:
  • the beginning of the sequence
  • its name or description
  • any references or links
  • any formulas
  • cross-references to other sequences
  • the name of the person who submitted it, etc.
  For further information about the format of replies received from the database, click here. A second file describes the internal format in which the sequences are stored in the database. See also the hints file for further useful information.

• OEIS: The movie: To celebrate the launching of The OEIS Foundation, Tony Noe has made an 8.5-minute movie showing 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.
  • Going to Tony'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).

• OEIS: Brief history. The sequence database was begun by Neil J. A. Sloane (henceforth, "NJAS") in 1965 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 On-line Encyclopedia of Integer Sequences), 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. 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).

  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 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. (Incidentally, Simon Plouffe is now one of the Trustees of The OEIS Foundation Inc.) The 1973 book contained 2372 sequences, and the 1995 book 5487 sequences, occupying 587 pages.

  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 10,000 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 Web Site.

  During this period, from 1996 to 2009, the database grew by at least 10,000 entries per year. If it were to be published in book form today, it 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 dtabase 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 the database was moved from NJAS's home page at AT&T to a commercial hosting service. Two versions of the database are now available: one, the "classic" OEIS, is a continuation of the version that was on NJAS's home page; the other (not quite finished, as of June 2010) is a "moderated Wiki".

  For further information about The OEIS Foundation Inc., please see the Foundation's web site.

• Arrangement of Sequences in Database. Most of the sequences are arranged in the database in lexicographic order of absolute values, 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 strict 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 Sending in a new sequence.) Sequences received in the last few days and not yet placed in the lexicographic ordering will be found at the end of the table.

• Format used in replies from the database. Internal format used in the database.

• Short 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.

• The Recent Additions file. (Note that you can also browse these using the WebCam.)

• A gzipped file containing just the sequences and their A-numbers (about 9 megs)

• A gzipped file containing just the names of the sequences and their A-numbers (about 3 megs)

• Contributing a new sequence (or a comment on an existing sequence, or more terms for an existing sequence).

  Want to help? Set the WebCam to browse the sequences that need extending,
  or use the main look-up page to search for keyword:more.
  See also the future projects web page.
  Other related pages: Demos, Transformations of sequences, Maple or Mathematica (see EISFormat.m) scripts to format sequences.

• OEIS Search Bar. To add OEIS search to the search bar in Firefox (1.5 or later), Internet Explorer (7 or later), or Mozilla Seamonkey (and perhaps Camino) click here.
  To add OEIS search to the search bar in Opera (9 or later), go to the main lookup page, right click in the text box, and select "Create search...". You can also add the OEIS to the search bar using Tools -> Preferences -> Search.
  It may be possible to add OEIS search to Safary by using the Saft plugin.
  There are also OpenSearch and Sherlock OEIS search plugins available at mycroft.mozdev.org
  (Thanks to several sequence fans who set up these plugins.)
• Sequences in Classic Books. Comtet's Advanced Combinatorics, Flajolet and Sedgewick's Analytic Combinatorics, Graham, Knuth and Patashnik's Concrete Mathematics, Harary and Palmer's Graphical Enumeration, Stanley's Enumerative Combinatorics.
• 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/classic/index.html.

  The URL for the present page is

  http://oeis.org/classic/Seis.html.
• 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 database.

  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://eis.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]).

• URL for Searching the Database
  To bypass the web page and search for a sequence directly using the cgi program, for instance the sequence 2,5,14,50,233,
  use (with no line break and no internal spaces):

      http://www.research.att.com/~njas/sequences/index.html?q=2,5,14,50,233
  

  To put a window on your own page to do lookups, use the following html commands:

      To look up a number sequence in the
      <a href="http://www.research.att.com/~njas/sequences/">
      On-Line Encyclopedia of Integer Sequences</a>,
      enter it here and click "Submit":
      <form
      action="http://www.research.att.com/~njas/sequences/"
      method=get>
      <input type=text name=q SIZE=60 VALUE=
      "1,2,3,6,11,23,47,106,235">
      <input type=submit VALUE="Submit">
      </form>
  

  To bypass the web page and search for a word or phrase directly using the cgi program, for instance the phrase "number of factors",
  use (with no line break and no internal spaces):

      http://www.research.att.com/~njas/sequences/index.html?q="number of factors"
  

• 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.

• Policy on Email Addresses in the OEIS
  If possible we prefer to give the author's name and email address with each sequence, so that people can get in touch with each other. This is an important feature of the database.

  Email addresses are disguised by replacing @ by (AT).

  Let one of the editors know if you don't want your email address to appear in any form. However, if you do ask to have your email address removed, try to provide a link to your home page, something like this:

  %H A077001 John Smith, <a href="http://members.aol.org/~JSmth/">Home Page</a> (listed in lieu of email address)

  that we can add to each of your sequences.

  Again, when sending in a sequence or comment using the Contribute new seq. or comment web page, if you don't want your email address to be used, say so in one of the windows, and if possible put the URL of your home page into one of the "links" windows.

• Copyright Notice. This database and its associated files are copyright 2010 by The OEIS Foundation Inc.
• 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 are due 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":

  • A007318 (Pascal's triangle),
  • A008277 (triangle of Stirling numbers of second kind),
  • A011971 (Aitken's array),
  • A026300 (Motzkin's triangle),
  • A034851 (Losanitsch's triangle).

  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. The first versions of the new programs were written by Alex Healy and the final versions by Russ Cox. My colleague David Applegate then helped install them on our new server. The new searches are much faster than the old ones and can handle much more complicated queries. See the hints file for details.

• Links.
  • 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
  • Weisstein's MathWorld
  • Neil Sloane's home page has many more links.

• Press Clippings, Awards, etc. The OEIS has often been written about by the press - see the Press Clippings section of NJAS's home page. Also:

  (Dec 05, 2007)	(May 10, 2001)	(Mar 09, 2000)
  (Jun 12, 2000)	(May 15, 1998)	(Apr 29, 1997)	(1997)

  (May 22, 1997)

  (Oct. 9, 1996)

  (1996)

Click the single right arrow to go to the next demonstration page, or the single left arrow to return to the previous page.