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<link rel="EditURI" type="application/rsd+xml" href="https://oeis.org/w/api.php?action=rsd"/> <link rel="license" href="/wiki/The_OEIS_End-User_License_Agreement"/> <link rel="alternate" type="application/atom+xml" title="OeisWiki Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"/>  </head> <body class="mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Welcome rootpage-Welcome skin-monobook action-view"><div id="globalWrapper"> <div id="column-content"> <div id="content" class="mw-body" role="main"> <a id="top"></a> <div id="siteNotice" class="mw-body-content"><div id="localNotice" lang="en" dir="ltr"><div class="mw-parser-output"><p>This site is supported by donations to <a rel="nofollow" class="external text" href="http://oeisf.org/">The OEIS Foundation</a>. </p></div></div></div> <div class="mw-indicators mw-body-content"> </div> <h1 id="firstHeading" class="firstHeading" lang="en">Welcome</h1> <div id="bodyContent" class="mw-body-content"> <div id="siteSub">From OeisWiki</div> <div id="contentSub"><div id="mw-fr-revisiontag" class="flaggedrevs_quality flaggedrevs_stable_synced plainlinks noprint nomobile"><img class="flaggedrevs-icon" src="/w/extensions/FlaggedRevs/frontend/modules/img/3.png" alt="This is a quality version of this page" title="This is a quality version of this page" />This is the <a href="/wiki/Help:Page_validation" title="Help:Page validation">stable version</a>, <a rel="nofollow" class="external text" href="https://oeis.org/w/index.php?title=Special:Log&type=review&page=Welcome">approved</a> on <i>16 October 2024</i>.<a id="mw-fr-revisiontoggle" class="fr-toggle-symbol" style="display:none;" title="show/hide details" >(+)</a><div id='mw-fr-revisiondetails'> <span class='fr-marker-60'><strong>Readability</strong>: <span class='fr-text-value'>Approved </span> </span> </div></div> </div> <div id="jump-to-nav" class="mw-jump">Jump to: <a href="#column-one">navigation</a>, <a href="#searchInput">search</a></div>  <div id="mw-content-text" lang="en" dir="ltr" class="mw-content-ltr"><div class="mw-parser-output"><p><br /> NOTE: The <a href="/wiki/Main_Page" title="Main Page"><b>Main Page</b></a> on the OEIS Wiki has much more information (FAQ, Index, Style Sheet, Trouble Logging In, Citations, etc.) </p><p><br /> </p> <div id="toc" class="toc"><div class="toctitle"><h2>Contents</h2></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Welcome_to_The_On-Line_Encyclopedia_of_Integer_Sequences.C2.AE_.28OEIS.C2.AE.29_Wiki"><span class="tocnumber">1</span> <span class="toctext">Welcome to The On-Line Encyclopedia of Integer Sequences® (OEIS®) Wiki</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Some_Famous_Sequences"><span class="tocnumber">2</span> <span class="toctext">Some Famous Sequences</span></a></li> <li class="toclevel-1 tocsection-3"><a href="#General_Information_About_OEIS"><span class="tocnumber">3</span> <span class="toctext">General Information About OEIS</span></a></li> <li class="toclevel-1 tocsection-4"><a href="#Introductory_chapters_from_the_1973_and_1995_books.3B_Supplement_3_to_1973_book"><span class="tocnumber">4</span> <span class="toctext">Introductory chapters from the 1973 and 1995 books; Supplement 3 to 1973 book</span></a></li> <li class="toclevel-1 tocsection-5"><a href="#Description_of_OEIS_entries_.28or.2C_What_is_the_Next_Term.3F.29"><span class="tocnumber">5</span> <span class="toctext">Description of OEIS entries (or, What is the Next Term?)</span></a></li> <li class="toclevel-1 tocsection-6"><a href="#OEIS:_Brief_History"><span class="tocnumber">6</span> <span class="toctext">OEIS: Brief History</span></a></li> <li class="toclevel-1 tocsection-7"><a href="#OEIS:_The_Movie"><span class="tocnumber">7</span> <span class="toctext">OEIS: The Movie</span></a></li> <li class="toclevel-1 tocsection-8"><a href="#Arrangement_of_the_Sequences_in_Database"><span class="tocnumber">8</span> <span class="toctext">Arrangement of the Sequences in Database</span></a></li> <li class="toclevel-1 tocsection-9"><a href="#Format_Used_in_Replies_From_the_Database"><span class="tocnumber">9</span> <span class="toctext">Format Used in Replies From the Database</span></a></li> <li class="toclevel-1 tocsection-10"><a href="#Index"><span class="tocnumber">10</span> <span class="toctext">Index</span></a></li> <li class="toclevel-1 tocsection-11"><a href="#Sequences_Which_Agree_For_a_Long_Time"><span class="tocnumber">11</span> <span class="toctext">Sequences Which Agree For a Long Time</span></a></li> <li class="toclevel-1 tocsection-12"><a href="#Recent_Additions"><span class="tocnumber">12</span> <span class="toctext">Recent Additions</span></a></li> <li class="toclevel-1 tocsection-13"><a href="#Compressed_Versions"><span class="tocnumber">13</span> <span class="toctext">Compressed Versions</span></a></li> <li class="toclevel-1 tocsection-14"><a href="#Contributing_a_New_Sequence.2C_Comment_or_More_Terms"><span class="tocnumber">14</span> <span class="toctext">Contributing a New Sequence, Comment or More Terms</span></a></li> <li class="toclevel-1 tocsection-15"><a href="#OEIS_Search_Bar"><span class="tocnumber">15</span> <span class="toctext">OEIS Search Bar</span></a></li> <li class="toclevel-1 tocsection-16"><a href="#Email_Addresses.2C_Getting_in_Touch_With_Authors"><span class="tocnumber">16</span> <span class="toctext">Email Addresses, Getting in Touch With Authors</span></a></li> <li class="toclevel-1 tocsection-17"><a href="#Ombudsman"><span class="tocnumber">17</span> <span class="toctext">Ombudsman</span></a></li> <li class="toclevel-1 tocsection-18"><a href="#Sequences_in_Classic_Books"><span class="tocnumber">18</span> <span class="toctext">Sequences in Classic Books</span></a></li> <li class="toclevel-1 tocsection-19"><a href="#Citations"><span class="tocnumber">19</span> <span class="toctext">Citations</span></a></li> <li class="toclevel-1 tocsection-20"><a href="#Referencing_the_OEIS"><span class="tocnumber">20</span> <span class="toctext">Referencing the OEIS</span></a></li> <li class="toclevel-1 tocsection-21"><a href="#URLs"><span class="tocnumber">21</span> <span class="toctext">URLs</span></a></li> <li class="toclevel-1 tocsection-22"><a href="#Referencing_a_Particular_Sequence"><span class="tocnumber">22</span> <span class="toctext">Referencing a Particular Sequence</span></a></li> <li class="toclevel-1 tocsection-23"><a href="#Acknowledgments"><span class="tocnumber">23</span> <span class="toctext">Acknowledgments</span></a></li> <li class="toclevel-1 tocsection-24"><a href="#Like_us_on_Facebook.21"><span class="tocnumber">24</span> <span class="toctext">Like us on Facebook!</span></a></li> <li class="toclevel-1 tocsection-25"><a href="#Links_to_Other_Sites"><span class="tocnumber">25</span> <span class="toctext">Links to Other Sites</span></a></li> <li class="toclevel-1 tocsection-26"><a href="#OEIS_Mentioned_in_WolframAlpha_Timeline"><span class="tocnumber">26</span> <span class="toctext">OEIS Mentioned in WolframAlpha Timeline</span></a></li> <li class="toclevel-1 tocsection-27"><a href="#Awards_and_press_clippings"><span class="tocnumber">27</span> <span class="toctext">Awards and press clippings</span></a></li> <li class="toclevel-1 tocsection-28"><a href="#Copyright_Notice"><span class="tocnumber">28</span> <span class="toctext">Copyright Notice</span></a></li> </ul> </div> <h2><span class="mw-headline" id="Welcome_to_The_On-Line_Encyclopedia_of_Integer_Sequences.C2.AE_.28OEIS.C2.AE.29_Wiki">Welcome to The On-Line Encyclopedia of Integer Sequences® (OEIS®) Wiki</span></h2> <h2><span class="mw-headline" id="Some_Famous_Sequences">Some Famous Sequences</span></h2> <p>Click on any of the following to see examples of famous sequences in the <a href="http://oeis.org/" class="extiw" title="oeis:"><b>On-Line Encyclopedia of Integer Sequences</b></a> (the <a href="http://oeis.org/" class="extiw" title="oeis:"><b>OEIS</b></a>), then hit "Back" in your browser to return here: </p> <ul><li> Recamán's sequence, <a href="http://oeis.org/A005132">A005132</a></li> <li> The Busy Beaver problem, <a href="http://oeis.org/A060843">A060843</a></li> <li> The Catalan numbers, <a href="http://oeis.org/A000108">A000108</a></li> <li> The prime numbers, <a href="http://oeis.org/A000040">A000040</a></li> <li> The Mersenne primes, <a href="http://oeis.org/A000043">A000043</a> and <a href="http://oeis.org/A000668">A000668</a></li> <li> The Fibonacci numbers, <a href="http://oeis.org/A000045">A000045</a></li></ul> <p>For some other fascinating sequences see <a rel="nofollow" class="external text" href="https://oeis.org/OEIS_pics.html"><b>Pictures from the OEIS: The (Free) OEIS Store</b></a> </p> <h2><span class="mw-headline" id="General_Information_About_OEIS">General Information About OEIS</span></h2> <ul><li> Most people use the <a href="http://oeis.org/" class="extiw" title="oeis:"><b>OEIS</b></a> 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 <a rel="nofollow" class="external text" href="https://oeis.org"><b>main look-up page</b></a>, enter the sequence, and click <b>Search</b>. You could also look for your sequence in the <a href="/wiki/Index_to_OEIS" title="Index to OEIS"> <b>Index</b></a>.</li></ul> <ul><li> You might also look at the <a rel="nofollow" class="external text" href="https://oeis.org/demo1.html"><b>demonstration pages</b></a> to see more examples of how to use the OEIS.</li></ul> <ul><li> If your favorite sequence isn't in the database, and if it is interesting, please submit it using the web page for <a rel="nofollow" class="external text" href="https://oeis.org/Submit.html"><b>Contributing a new sequence or comment</b></a>. 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 <a href="/wiki/Special:RequestAccount" title="Special:RequestAccount"><b>register</b></a>.</li></ul> <ul><li> If you have stumped the database, you can try <b>Superseeker</b>, which tries really hard to identify a sequence. Send an empty email message to <b>superseeker@oeis.org</b> to get instructions. </li></ul> <ul><li> You can browse the database, using the <a rel="nofollow" class="external text" href="https://oeis.org/webcam"><b>WebCam</b></a>. This can be set to look at the most interesting sequences, recent additions, or sequences needing more terms. It can be quite addictive!</li></ul> <ul><li> It is also interesting to browse the <a href="/wiki/Index_to_OEIS" title="Index to OEIS"> <b>Index</b></a> 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.</li></ul> <ul><li> Also worth visiting are the pages dealing with <a rel="nofollow" class="external text" href="https://oeis.org/Spuzzle.html"><b>Puzzle sequences</b></a>, <a rel="nofollow" class="external text" href="https://oeis.org/classic.html"><b>Classic sequences</b></a> and <a rel="nofollow" class="external text" href="https://oeis.org/Shot.html"><b>Hot sequences</b></a>.</li></ul> <ul><li> You might also like to see the <a href="/wiki/Works_Citing_OEIS" title="Works Citing OEIS"> <b>list of papers that have acknowledged help from the database</b></a> and some <a rel="nofollow" class="external text" href="https://oeis.org/demol.html"><b>comments from readers</b></a>.</li></ul> <h2><span class="mw-headline" id="Introductory_chapters_from_the_1973_and_1995_books.3B_Supplement_3_to_1973_book">Introductory chapters from the 1973 and 1995 books; Supplement 3 to 1973 book</span></h2> <p>The introductory chapters from N. J. A. Sloane's <b>Handbook of Integer Sequences</b> (1973) (click <a rel="nofollow" class="external text" href="https://oeis.org/HIStext.pdf">here</a>) and N. J. A. Sloane and S. Plouffe's <b>Encyclopedia of Integer Sequences</b> (1995) (click <a rel="nofollow" class="external text" href="https://oeis.org/EIStext.pdf">here</a>) contain much useful information about analyzing an unknown sequence and many other topics. </p><p>The third Supplement to the 1973 Handbook (which supersedes the first two supplements) is available here: <a href="/w/images/6/6d/Supp3.txt" class="internal" title="Supp3.txt">Media:Supp3.txt</a> </p> <h2><span class="mw-headline" id="Description_of_OEIS_entries_.28or.2C_What_is_the_Next_Term.3F.29">Description of OEIS entries (or, What is the Next Term?)</span></h2> <p><font color="red"><b>What comes next after 1, 2, 4, 9, 20, 48, 115, 286, 719, ... (for example)? This is the place to find out!</b></font> (Answer: click <a rel="nofollow" class="external text" href="https://oeis.org/A000081"><b>here</b></a>.) </p> <ul><li> 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: <ul><li> the beginning of the sequence</li> <li> its name or description</li> <li> a graph of the sequence</li> <li> additional comments</li> <li> the offset (index of first term)</li> <li> references or links</li> <li> formulas</li> <li> computer programs</li> <li> cross-references to other sequences</li> <li> the sequence converted to music</li> <li> the name of the person who submitted it</li> <li> the history of the OEIS entry </li></ul></li></ul> <ul><li> For further information about the format of replies received from the database, <a rel="nofollow" class="external text" href="https://oeis.org/eishelp2.html"><b>click here</b></a>. See also the <a href="/wiki/Hints" class="mw-redirect" title="Hints"><b>hints file</b></a> for further useful information.</li></ul> <ul><li> The sequence pages have little buttons at the top marked "<b>list</b>", "<b>graph</b>", "<b>refs</b>", "<b>listen</b>", "<b>history</b>", "<b>text</b>", "<b>internal format</b>", and sometimes "<b>table</b>" and "<b>edit</b>". <ul><li> "<b>list</b>" produces a numbered list of the terms, plus a bracketed list suitable for importing into other programs such as Maple, Mathemtica, PARI, etc.</li> <li> "<b>refs</b>" shows all the sequences which reference this one.</li> <li> "<b>table</b>": 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, <a href="http://oeis.org/A007318">A007318</a>.</li> <li> "<b>graph</b>" 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 <a href="http://oeis.org/A000045">A000045</a>, the partition numbers <a href="http://oeis.org/A000041">A000041</a>, the Euler phi-function <a href="http://oeis.org/A000010">A000010</a>, etc.</li> <li> "<b>listen</b>" 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 <a href="http://oeis.org/A005132">A005132</a>, turn the volume up to 127 and set the instrument to #103 !</li> <li> "<b>history</b>" shows the history of changes to this entry.</li> <li> "<b>edit</b>" (you will see this only if you are a <a href="/wiki/Special:RequestAccount" title="Special:RequestAccount"><b>registered</b></a> user) enables you to propose changes to the sequence.</li> <li> "<b>text</b>" shows the various sections of the entry for this sequence with each line prefixed by a symbol that indicates its function.</li> <li> "<b>Internal format</b>" shows the various sections of the entry for this sequence in the internal format.</li></ul></li></ul> <h2><span class="mw-headline" id="OEIS:_Brief_History">OEIS: Brief History</span></h2> <p>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 <a href="http://oeis.org/A000435">A000435</a> 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. </p><p>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. </p><p>Here is a scan of the <a rel="nofollow" class="external text" href="https://oeis.org/A000435/a000435_1.pdf"><b>page</b></a> in NJAS's thesis <a rel="nofollow" class="external text" href="https://oeis.org/A000435/a000435_2.pdf"><b>notebook</b></a> with the very first collection of sequences. (The sequences mentioned are <a href="http://oeis.org/A000027">A000027</a>, <a href="http://oeis.org/A000217">A000217</a>, <a href="http://oeis.org/A000292">A000292</a>, <a href="http://oeis.org/A000332">A000332</a>, <a href="http://oeis.org/A000389">A000389</a>, <a href="http://oeis.org/A000579">A000579</a>, <a href="http://oeis.org/A000110">A000110</a>, <a href="http://oeis.org/A007318">A007318</a>, <a href="http://oeis.org/A000058">A000058</a>, <a href="http://oeis.org/A000215">A000215</a>, <a href="http://oeis.org/A000289">A000289</a>, <a href="http://oeis.org/A000324">A000324</a>, <a href="http://oeis.org/A234953">A234953</a> (= <a href="http://oeis.org/A001854">A001854</a>(n)/n), <a href="http://oeis.org/A000435">A000435</a>, <a href="http://oeis.org/A000169">A000169</a>, <a href="http://oeis.org/A000142">A000142</a>, <a href="http://oeis.org/A000272">A000272</a>, <a href="http://oeis.org/A000312">A000312</a>, <a href="http://oeis.org/A000111">A000111</a>.) This is the acorn from which the OEIS grew. The date is January or February, 1964. </p><p>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. </p><p>[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".] </p><p>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. </p><p>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 <a rel="nofollow" class="external text" href="https://oeisf.org">The OEIS Foundation Inc.</a>.) </p><p>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. </p><p>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. </p><p>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. </p><p>In 2009, therefore, it was decided to make a drastic change. NJAS set up a non-profit foundation, <a rel="nofollow" class="external text" href="https://oeisf.org"><b>The OEIS Foundation Inc.</b></a>, 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. </p><p>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. </p><p>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. </p><p>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. </p> <h2><span class="mw-headline" id="OEIS:_The_Movie">OEIS: The Movie</span></h2> <p>To celebrate the launching of <a rel="nofollow" class="external text" href="https://oeisf.org"><b>The OEIS Foundation Inc</b></a>, Tony Noe made an 8.5-minute movie showing graphs of the first 1000 terms of 1000 sequences, with soundtrack from Recaman's sequence <a href="http://oeis.org/A005132">A005132</a>. </p><p>There are four ways to view the movie: </p> <ul><li> On <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=LCWglXljevY">YouTube</a> (you can find it by searching for "OEIS" and "Movie").</li> <li> By downloading a <a rel="nofollow" class="external text" href="http://www.sspectra.com/math/OEISMovie.mov">5 MB QuickTime movie</a> that is viewable with QuickTime Player 7 and some browsers.</li> <li> By downloading a <a rel="nofollow" class="external text" href="http://www.sspectra.com/math/OEISMovie2.mov">27 MB movie</a> 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.</li> <li> By going to <a rel="nofollow" class="external text" href="http://www.sspectra.com/math/MovieFrames/">Tony Noe's website</a> for a frame-by-frame display, with links to the definitions of the sequences.</li></ul> <h2><span class="mw-headline" id="Arrangement_of_the_Sequences_in_Database">Arrangement of the Sequences in Database</span></h2> <ul><li> Sequences in the OEIS are arranged in <b>lexicographic order</b>, 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.</li></ul> <ul><li> 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 <a rel="nofollow" class="external text" href="https://oeis.org/Submit.html"><b>contributing a new sequence or comment</b></a>.)</li></ul> <ul><li> Each entry in the OEIS has a link called <b>Sequence in context</b>, 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.)</li></ul> <ul><li> There is also a link called <b>Adjacent sequences</b>, which shows the three entries whose A-numbers are immediately before and after the current sequence.</li></ul> <ul><li> 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.</li></ul> <h2><span class="mw-headline" id="Format_Used_in_Replies_From_the_Database">Format Used in Replies From the Database</span></h2> <p>For information about the format of replies received from the database, <a rel="nofollow" class="external text" href="https://oeis.org/eishelp2.html"><b>click here</b></a>. See also the <a href="/wiki/Hints" class="mw-redirect" title="Hints"><b>hints file</b></a> for further useful information. </p> <h2><span class="mw-headline" id="Index">Index</span></h2> <ul><li> There is an <a href="/wiki/Index_to_OEIS" title="Index to OEIS"> <b>Index</b></a> to the most important sequences. </li> <li> The <b><a href="/wiki/Main_Page" title="Main Page">main look-up page</a></b> will also allow you to search for a word (or do much more complicated searches) in the database.</li></ul> <h2><span class="mw-headline" id="Sequences_Which_Agree_For_a_Long_Time">Sequences Which Agree For a Long Time</span></h2> <ul><li> People are always asking about this, so there is a section about them in the <a href="/wiki/Index_to_OEIS:_Section_Se#sequences_which_agree_for_a_long_time" title="Index to OEIS: Section Se"> <b>Index</b></a>.</li></ul> <h2><span class="mw-headline" id="Recent_Additions">Recent Additions</span></h2> <ul><li> Recent additions to the OEIS can be seen by clicking the <a rel="nofollow" class="external text" href="https://oeis.org/recent"><b>Recent Additions</b></a> link. </li> <li> You can also browse the recent additions using the <a rel="nofollow" class="external text" href="https://oeis.org/webcam"><b>WebCam</b></a>.</li></ul> <h2><span class="mw-headline" id="Compressed_Versions">Compressed Versions</span></h2> <ul><li> There is a <a rel="nofollow" class="external text" href="https://oeis.org/stripped.gz"><b>gzipped file</b></a> containing just the sequences and their A-numbers (a few tens of megabytes)</li> <li> There is also a <a rel="nofollow" class="external text" href="https://oeis.org/names.gz"><b>gzipped file</b></a> containing just the names of the sequences and their A-numbers (a few megabytes)</li></ul> <p>These two files are updated daily. </p> <h2><span class="mw-headline" id="Contributing_a_New_Sequence.2C_Comment_or_More_Terms">Contributing a New Sequence, Comment or More Terms</span></h2> <ul><li> For contributing a new sequence, comment or more terms for an existing sequence, see <a rel="nofollow" class="external text" href="https://oeis.org/Submit.html"><b>this page</b></a>.</li> <li> Want to help? <ul><li> See the web page <a rel="nofollow" class="external text" href="https://oeis.org/more.html"><b>Sequences that need more terms</b></a> </li> <li> Or use the <a rel="nofollow" class="external text" href="https://oeis.org/webcam"><b>WebCam</b></a> to browse the sequences that need more terms </li> <li> Or use the <a rel="nofollow" class="external text" href="https://oeis.org"><b>main look-up page</b></a> to search for the keyword <b>more</b> </li></ul></li></ul> <ul><li> See also the page <a rel="nofollow" class="external text" href="https://oeis.org/classic/future.html"><b>future projects</b></a> (this page needs updating)</li></ul> <h2><span class="mw-headline" id="OEIS_Search_Bar">OEIS Search Bar</span></h2> <p>To add an OEIS search bar to your browser, see the instructions <a rel="nofollow" class="external text" href="https://oeis.org/Searchbar.html"><b>here</b></a>. </p> <h2><span class="mw-headline" id="Email_Addresses.2C_Getting_in_Touch_With_Authors">Email Addresses, Getting in Touch With Authors</span></h2> <p>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. </p><p>First, find the author in the list of <a href="/wiki/Special:ListUsers" title="Special:ListUsers"><b>Registered Users</b></a> 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". </p> <h2><span class="mw-headline" id="Ombudsman">Ombudsman</span></h2> <p><b>Hilarie Orman</b> has volunteered to serve as an <b>ombudsman</b> to help resolve disputes with contributors. She can be reached through her <a href="/wiki/User:Hilarie_Orman" title="User:Hilarie Orman">user page</a> on this wiki. </p> <h2><span class="mw-headline" id="Sequences_in_Classic_Books">Sequences in Classic Books</span></h2> <p>This page has moved. Please go to <a href="/wiki/Sequences_From_Classic_Books" title="Sequences From Classic Books">Sequences From Classic Books</a>. </p> <h2><span class="mw-headline" id="Citations">Citations</span></h2> <ul><li> The lists of <a href="/wiki/Works_Citing_OEIS" title="Works Citing OEIS"><b>Works Citing the Encyclopedia of Integer Sequences</b></a> show some of the ways that people have used the database.</li></ul> <h2><span class="mw-headline" id="Referencing_the_OEIS">Referencing the OEIS</span></h2> <p>If you have found the <a href="http://oeis.org/" class="extiw" title="oeis:"><b>OEIS</b></a> useful and wish to reference it, the usual citation is </p> <dl><dd>OEIS Foundation Inc. (2024), The On-Line Encyclopedia of Integer Sequences, Published electronically at <a rel="nofollow" class="external free" href="https://oeis.org">https://oeis.org</a>.</dd></dl> <h2><span class="mw-headline" id="URLs">URLs</span></h2> <ul><li> The URL for the main lookup page is <a href="/wiki/Main_Page" title="Main Page"><b>https://oeis.org</b></a></li></ul> <ul><li> The URL for the present page is <a class="mw-selflink selflink"><b>https://oeis.org/wiki/Welcome</b></a></li></ul> <h2><span class="mw-headline" id="Referencing_a_Particular_Sequence">Referencing a Particular Sequence</span></h2> <ul><li> If you are writing a paper and wish to refer the Catalan numbers, say (sequence <a href="http://oeis.org/A000108">A000108</a>), but don't want to digress to describe them, simply add a reference or link that points directly to that sequence in the OEIS.</li></ul> <ul><li> The URL for sequence <a href="http://oeis.org/A000108">A000108</a> (for example) is <b><a rel="nofollow" class="external free" href="https://oeis.org/A000108">https://oeis.org/A000108</a></b>.</li></ul> <ul><li> A text reference might say:</li></ul> <p><i>The On-Line Encyclopedia of Integer Sequences</i>, published electronically at <a rel="nofollow" class="external free" href="https://oeis.org">https://oeis.org</a>, 2010, Sequence <a href="http://oeis.org/A000108">A000108</a> </p><p>or, if it is clear who "discovered" the sequence, something like </p><p>J. H. Conway, Sequence <a href="http://oeis.org/A007970">A007970</a> in <i>The On-Line Encyclopedia of Integer Sequences</i> (2010), published electronically at <a rel="nofollow" class="external free" href="https://oeis.org">https://oeis.org</a>. </p> <ul><li> In an HTML file one might say something like this: ... where the C(n) are the Catalan numbers (<a href="https://oeis.org/<a href="http://oeis.org/A000108">A000108</a>">Sequence <a href="http://oeis.org/A000108">A000108</a></a> in [OEIS]).</li></ul> <h2><span class="mw-headline" id="Acknowledgments">Acknowledgments</span></h2> <ul><li> A very large number of people have contributed to the database, and it would be impossible to thank them individually. </li></ul> <ul><li> Special thanks to <a rel="nofollow" class="external text" href="http://www.iki.fi/~kartturi/">Antti Karttunen</a>, 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": <ul><li> <a href="http://oeis.org/A007318">A007318</a> (Pascal's triangle),</li> <li> <a href="http://oeis.org/A008277">A008277</a> (triangle of Stirling numbers of second kind),</li> <li> <a href="http://oeis.org/A011971">A011971</a> (Aitken's array),</li> <li> <a href="http://oeis.org/A026300">A026300</a> (Motzkin's triangle),</li> <li> <a href="http://oeis.org/A034851">A034851</a> (Losanitsch's triangle). </li></ul></li></ul> <h2><span class="mw-headline" id="Like_us_on_Facebook.21">Like us on Facebook!</span></h2> <p>See <a rel="nofollow" class="external text" href="https://www.facebook.com/OEISF">here</a> </p> <h2><span class="mw-headline" id="Links_to_Other_Sites">Links to Other Sites</span></h2> <ul><li> <a rel="nofollow" class="external text" href="https://en.wikipedia.org/wiki/ATLAS_of_Finite_Groups"><b>ATLAS of Finite Groups</b></a>. The original ATLAS of Finite Groups, that is, J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups, Oxford Univ. Press, 1985, seems (in 2024) to be no longer available online. The closest replacement for it is the <a rel="nofollow" class="external text" href="https://brauer.maths.qmul.ac.uk/Atlas/"><b>ATLAS of Finite Group Representations</b></a> by Robert A. Wilson and numerous coauthors.</li> <li> <a rel="nofollow" class="external text" href="https://primes.utm.edu/"><b>Caldwell's Prime Pages</b></a></li> <li> <a rel="nofollow" class="external text" href="http://combos.org/"><b>Combinatorial Object Server</b></a> (originally by Frank Ruskey)</li> <li> <a rel="nofollow" class="external text" href="http://jeux-et-mathematiques.davalan.org/"><b>Davalan's Jeux et Mathématiques</b></a></li> <li> <a rel="nofollow" class="external text" href="http://www.worldofnumbers.com/"><b>De Geest's World of Numbers</b></a></li> <li> <a rel="nofollow" class="external text" href="https://github.com/jlumbroso/encyclopedia-of-combinatorial-structures-data"><b>Encyclopedia of Combinatorial Structures</b></a></li> <li> <b>Mathematical Constants</b> by Steven R. Finch <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" class="extiw" title="wikipedia:Digital object identifier"><span style="border-bottom: 1px dotted #999" title="digital object identifier">doi</span></a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2F9781316997741">10.1017/9781316997741</a></li> <li> <a rel="nofollow" class="external text" href="https://www.ics.uci.edu/~eppstein/junkyard/"><b>Geometry Junkyard</b></a></li> <li> <a rel="nofollow" class="external text" href="https://cs.uwaterloo.ca/journals/JIS/"><b>Journal of Integer Sequences</b></a></li> <li> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet/"><b>MathSciNet</b></a></li> <li> <a rel="nofollow" class="external text" href="https://primes.utm.edu/nthprime/"><b>The Nth Prime Page</b></a></li> <li> <a href="/wiki/Plouffe%27s_Inverter" title="Plouffe's Inverter"><b>Plouffe's Inverter</b></a> (see also the <a rel="nofollow" class="external text" href="http://wayback.cecm.sfu.ca/projects/ISC/ISCmain.html"><b>Inverse Symbolic Calculator</b></a>)</li> <li> <a rel="nofollow" class="external text" href="https://groups.google.com/g/seqfan"><b>SeqFan Google Group</b></a></li> <li> <a rel="nofollow" class="external text" href="https://www.ellipsa.eu/public/primo/primo.html"><b>Primo</b></a> (primality proving program)</li> <li> <a rel="nofollow" class="external text" href="http://neilsloane.com/"><b>Neil Sloane's home page</b></a></li> <li> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/"><b>Eric Weisstein's MathWorld</b></a></li> <li> Software <ul><li> <a href="/wiki/User:R._J._Mathar/transforms3" title="User:R. J. Mathar/transforms3">b-file Maple programs written by R. J. Mathar</a></li> <li> <a rel="nofollow" class="external text" href="https://github.com/archmageirvine/joeis">Java OEIS written by S. A. Irvine and G. Fischer</a></li></ul></li></ul> <h2><span class="mw-headline" id="OEIS_Mentioned_in_WolframAlpha_Timeline">OEIS Mentioned in WolframAlpha Timeline</span></h2> <ul><li> The OEIS appears (under 1973) in WolframAlpha's <a rel="nofollow" class="external text" href="https://www.wolframalpha.com/docs/timeline/computable-knowledge-history-6.html">Timeline of Systematic Data and the Development of Computable Knowledge</a>, which extends from cave paintings to the present day.</li></ul> <h2><span class="mw-headline" id="Awards_and_press_clippings">Awards and press clippings</span></h2> <p>See the separate <a href="/wiki/Awards_and_press_clippings" title="Awards and press clippings">Awards and press clippings</a> page for the full list back to the 1970s. </p> <ul><li> The TV series <a rel="nofollow" class="external text" href="https://www.imdb.com/title/tt4158110/"><b>Mr Robot</b></a> 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).</li> <li> The Dutch magazine <a rel="nofollow" class="external text" href="https://www.pyth.eu/"><b>Pythagoras</b></a> 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).</li> <li> Margaret Wertheim, <a rel="nofollow" class="external text" href="https://oeis.org/Cabinet.pdf"><b>The Fax Numbers of the Beast, and Other Mathematical Sports: An Interview with Neil Sloane</b></a>, <a rel="nofollow" class="external text" href="https://www.cabinetmagazine.org/"><b>Cabinet Magazine</b></a>, Issue 57, Spring 2015, pages 48-54.</li> <li> Siobhan Roberts, <a rel="nofollow" class="external text" href="https://nautil.us/issue/29/scaling/how-to-build-a-search-engine-for-mathematics"><b>How to Build a Search Engine for Mathematics: The surprising power of Neil Sloane's Encyclopedia of Integer Sequences</b></a>, Nautilus Magazine, Issue 29, Chapter 4, Oct 22 2015.</li> <li> <b>Eric Egge</b> in <i>Defying God: The Stanley-Wilf Conjecture, Stanley-Wilf Limits, and a Two-Generation Explosion of Combinatorics</i>, pp. 65-82 of <i>A Century of Advancing Mathematic</i>, 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."</li> <li> Featured by <b><a rel="nofollow" class="external text" href="https://science.slashdot.org/story/15/08/07/021259/the-connoisseur-of-number-sequences">Slashdot</a></b> on August 7 2015</li> <li> Erica Klarreich, <a rel="nofollow" class="external text" href="https://www.quantamagazine.org/20150806-neil-sloane-oeis-interview/"><b>The Connoisseur of Number Sequences</b></a>, Quanta Magazine, August 6 2015 (Interview with Neil Sloane about the OEIS)</li> <li> <a rel="nofollow" class="external text" href="https://news.ycombinator.com/item?id=9919535"><b>Hacker News</b></a> discussion of OEIS, July 21 2015</li> <li> <a rel="nofollow" class="external text" href="https://oeiss.org/A000435/a000435.jpg"><b>Article by Vijayakumar Ambat in Malayalam</b></a> (a regional language of India) in the newspaper Malayala Manorama - Padhippura, 12 June 2015, that mentions the OEIS.</li> <li> <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=vKPRRUWbWAQ"><b>Video interview</b></a> about the OEIS by Marc Chamberland made at the Joint Math Meetings in San Antoio, Texas, Jan. 2015.</li> <li> Alex Bellos, <a rel="nofollow" class="external text" href="https://www.theguardian.com/science/alexs-adventures-in-numberland/2014/oct/07/neil-sloane-the-man-who-loved-only-integer-sequences"><b>Neil Sloane: the man who loved only integer sequences</b></a>, Alexs-adventures-in-numberland blog, The Guardian, Oct 07 2014.</li></ul> <h2><span class="mw-headline" id="Copyright_Notice">Copyright Notice</span></h2> <p>This database and its associated files are copyright 2024 by <a rel="nofollow" class="external text" href="https://oeisf.org">The OEIS Foundation Inc.</a>. </p>   </div>  </div><div class="printfooter"> Retrieved from "<a dir="ltr" href="https://oeis.org/w/index.php?title=Welcome&oldid=1661338">https://oeis.org/w/index.php?title=Welcome&oldid=1661338</a>"</div> <div id="catlinks" class="catlinks catlinks-allhidden" data-mw="interface"><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden category: <ul><li><a href="/wiki/Category:Pages_with_ignored_display_titles" title="Category:Pages with ignored display titles">Pages with ignored display titles</a></li></ul></div></div>  <div class="visualClear"></div> </div> </div> </div> <div id="column-one"> <h2>Navigation menu</h2> <div id="p-cactions" class="portlet" role="navigation"> <h3>Views</h3> <div class="pBody"> <ul> <li id="ca-nstab-main" class="selected"><a href="/wiki/Welcome" title="View the content page [c]" accesskey="c">Page</a></li> <li id="ca-talk"><a href="/wiki/Talk:Welcome" rel="discussion" title="Discussion about the content page [t]" accesskey="t">Discussion</a></li> <li id="ca-viewsource"><a href="/w/index.php?title=Welcome&action=edit" title="This page is protected.
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