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cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet menu__item--page-actions-watch"> Watch </a> </li> <li id="page-actions-edit" class="page-actions-menu__list-item"> <a role="button" id="ca-edit" href="/w/index.php?title=On-Line_Encyclopedia_of_Integer_Sequences&action=edit" data-event-name="menu.edit" data-mw="interface" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet edit-page menu__item--page-actions-edit"> Edit </a> </li> </ul> </nav>  <div id="mw-content-subtitle"></div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"><script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><section class="mf-section-0" id="mf-section-0"> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"OEIS" redirects here. For the birth defect known as OEIS complex, see <a href="/wiki/Cloacal_exstrophy" title="Cloacal exstrophy">Cloacal exstrophy</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style>The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of <a href="/wiki/Integer_sequence" title="Integer sequence">integer sequences</a>. It was created and maintained by <a href="/wiki/Neil_Sloane" title="Neil Sloane">Neil Sloane</a> while researching at <a href="/wiki/AT%26T_Labs" title="AT&T Labs">AT&T Labs</a>. He transferred the <a href="/wiki/Intellectual_property" title="Intellectual property">intellectual property</a> and hosting of the OEIS to the OEIS Foundation in 2009,<a href="#cite_note-4">[4]</a> and is its chairman. <table class="infobox vcard"><caption class="infobox-title fn org">On-Line Encyclopedia of Integer Sequences</caption><tbody><tr><td colspan="2" class="infobox-image"><a href="/wiki/File:OEIS_banner.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/OEIS_banner.png/300px-OEIS_banner.png" decoding="async" width="300" height="59" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/OEIS_banner.png/450px-OEIS_banner.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/OEIS_banner.png/600px-OEIS_banner.png 2x" data-file-width="629" data-file-height="124"></a></td></tr><tr><th scope="row" class="infobox-label">Founded</th><td class="infobox-data">1964; 60 years ago (1964)</td></tr><tr><th scope="row" class="infobox-label">Predecessor(s)</th><td class="infobox-data">Handbook of Integer Sequences, <a href="/wiki/Encyclopedia_of_Integer_Sequences" class="mw-redirect" title="Encyclopedia of Integer Sequences">Encyclopedia of Integer Sequences</a></td></tr><tr><th scope="row" class="infobox-label">Created by</th><td class="infobox-data"><a href="/wiki/Neil_Sloane" title="Neil Sloane">Neil Sloane</a></td></tr><tr><th scope="row" class="infobox-label">Chairman</th><td class="infobox-data"><a href="/wiki/Neil_Sloane" title="Neil Sloane">Neil Sloane</a></td></tr><tr><th scope="row" class="infobox-label">President</th><td class="infobox-data">Russ Cox</td></tr><tr><th scope="row" class="infobox-label">URL</th><td class="infobox-data url"><a rel="nofollow" class="external text" href="https://oeis.org/">oeis.org</a></td></tr><tr><th scope="row" class="infobox-label">Commercial</th><td class="infobox-data">No<a href="#cite_note-oeisfgoals-1">[1]</a></td></tr><tr><th scope="row" class="infobox-label">Registration</th><td class="infobox-data">Optional<a href="#cite_note-2">[2]</a></td></tr><tr><th scope="row" class="infobox-label">Launched</th><td class="infobox-data">1996; 28 years ago (1996)</td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Content license</div></th><td class="infobox-data"><a href="/wiki/Creative_Commons" title="Creative Commons">Creative Commons</a> <a href="/wiki/CC_BY-SA" class="mw-redirect" title="CC BY-SA">CC BY-SA</a> 4.0<a href="#cite_note-3">[3]</a></td></tr></tbody></table> OEIS records information on integer sequences of interest to both professional and <a href="/wiki/List_of_amateur_mathematicians" title="List of amateur mathematicians">amateur</a> <a href="/wiki/Mathematician" title="Mathematician">mathematicians</a>, and is widely cited. As of February 2024<a rel="nofollow" class="external text" href="https://oeis.org/">[ref]</a>, it contains over 370,000 sequences,<a href="#cite_note-5">[5]</a> and is growing by approximately 30 entries per day.<a href="#cite_note-6">[6]</a> Each entry contains the leading terms of the sequence, <a href="/wiki/Keyword_(computer_programming)" class="mw-redirect" title="Keyword (computer programming)">keywords</a>, mathematical motivations, literature links, and more, including the option to generate a <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph</a> or play a <a href="/wiki/Computer_music" title="Computer music">musical</a> representation of the sequence. The database is <a href="/wiki/Search_engine_(computing)" title="Search engine (computing)">searchable</a> by keyword, by <a href="/wiki/Subsequence" title="Subsequence">subsequence</a>, or by any of 16 fields. There is also an advanced search function called SuperSeeker which runs a large number of different algorithms to identify sequences related to the input.<a href="#cite_note-7">[7]</a> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><label class="toctogglelabel" for="toctogglecheckbox"></label></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#History">1 History</a></li> <li class="toclevel-1 tocsection-2"><a href="#Non-integers">2 Non-integers</a></li> <li class="toclevel-1 tocsection-3"><a href="#Conventions">3 Conventions</a> <ul> <li class="toclevel-2 tocsection-4"><a href="#Special_meaning_of_zero">3.1 Special meaning of zero</a></li> <li class="toclevel-2 tocsection-5"><a href="#Lexicographical_ordering">3.2 Lexicographical ordering</a></li> </ul> </li> <li class="toclevel-1 tocsection-6"><a href="#Self-referential_sequences">4 Self-referential sequences</a></li> <li class="toclevel-1 tocsection-7"><a href="#Abridged_example_of_a_comprehensive_entry">5 Abridged example of a comprehensive entry</a> <ul> <li class="toclevel-2 tocsection-8"><a href="#Entry_fields">5.1 Entry fields</a></li> </ul> </li> <li class="toclevel-1 tocsection-9"><a href="#Sloane's_gap">6 Sloane's gap</a></li> <li class="toclevel-1 tocsection-10"><a href="#See_also">7 See also</a></li> <li class="toclevel-1 tocsection-11"><a href="#Notes">8 Notes</a></li> <li class="toclevel-1 tocsection-12"><a href="#References">9 References</a></li> <li class="toclevel-1 tocsection-13"><a href="#Further_reading">10 Further reading</a></li> <li class="toclevel-1 tocsection-14"><a href="#External_links">11 External links</a></li> </ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"><h2 id="History">History</h2> <a role="button" href="/w/index.php?title=On-Line_Encyclopedia_of_Integer_Sequences&action=edit&section=1" title="Edit section: History" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-1 collapsible-block" id="mf-section-1"> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Encyclopedia_of_Integer_Sequences,_2nd_edition,_by_N.J.A._Sloane.jpg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/en/thumb/2/20/Encyclopedia_of_Integer_Sequences%2C_2nd_edition%2C_by_N.J.A._Sloane.jpg/150px-Encyclopedia_of_Integer_Sequences%2C_2nd_edition%2C_by_N.J.A._Sloane.jpg" decoding="async" width="150" height="226" class="mw-file-element" data-file-width="257" data-file-height="387"></noscript><span class="lazy-image-placeholder" style="width: 150px;height: 226px;" data-src="//upload.wikimedia.org/wikipedia/en/thumb/2/20/Encyclopedia_of_Integer_Sequences%2C_2nd_edition%2C_by_N.J.A._Sloane.jpg/150px-Encyclopedia_of_Integer_Sequences%2C_2nd_edition%2C_by_N.J.A._Sloane.jpg" data-width="150" data-height="226" data-srcset="//upload.wikimedia.org/wikipedia/en/thumb/2/20/Encyclopedia_of_Integer_Sequences%2C_2nd_edition%2C_by_N.J.A._Sloane.jpg/225px-Encyclopedia_of_Integer_Sequences%2C_2nd_edition%2C_by_N.J.A._Sloane.jpg 1.5x, //upload.wikimedia.org/wikipedia/en/2/20/Encyclopedia_of_Integer_Sequences%2C_2nd_edition%2C_by_N.J.A._Sloane.jpg 2x" data-class="mw-file-element"> </a><figcaption>Second edition of the book</figcaption></figure> <a href="/wiki/Neil_Sloane" title="Neil Sloane">Neil Sloane</a> started collecting integer sequences as a graduate student in 1964 to support his work in <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a>.<a href="#cite_note-8">[8]</a><a href="#cite_note-9">[9]</a> The database was at first stored on <a href="/wiki/Punched_card" title="Punched card">punched cards</a>. He published selections from the database in book form twice: <ol><li>A Handbook of Integer Sequences (1973, <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-648550-X" title="Special:BookSources/0-12-648550-X">0-12-648550-X</a>), containing 2,372 sequences in <a href="/wiki/Lexicographical_order" class="mw-redirect" title="Lexicographical order">lexicographic order</a> and assigned numbers from 1 to 2372.</li> <li>The Encyclopedia of Integer Sequences with <a href="/wiki/Simon_Plouffe" title="Simon Plouffe">Simon Plouffe</a> (1995, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-558630-2" title="Special:BookSources/0-12-558630-2">0-12-558630-2</a>), containing 5,488 sequences and assigned M-numbers from M0000 to M5487. The Encyclopedia includes the references to the corresponding sequences (which may differ in their few initial terms) in A Handbook of Integer Sequences as N-numbers from N0001 to N2372 (instead of 1 to 2372.) The Encyclopedia includes the A-numbers that are used in the OEIS, whereas the Handbook did not.</li></ol> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:OEIS-original_web_page.png" class="mw-file-description"><noscript><img alt='1999 "Integer Sequences" web page' src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/OEIS-original_web_page.png/220px-OEIS-original_web_page.png" decoding="async" width="220" height="189" class="mw-file-element" data-file-width="1017" data-file-height="875"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 189px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/OEIS-original_web_page.png/220px-OEIS-original_web_page.png" data-alt='1999 "Integer Sequences" web page' data-width="220" data-height="189" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/OEIS-original_web_page.png/330px-OEIS-original_web_page.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1a/OEIS-original_web_page.png/440px-OEIS-original_web_page.png 2x" data-class="mw-file-element"> </a><figcaption>Sloane's "Integer Sequences" web page on the "AT&T research" web site as of 1999</figcaption></figure> These books were well-received and, especially after the second publication, mathematicians supplied Sloane with a steady flow of new sequences. The collection became unmanageable in book form, and when the database had reached 16,000 entries Sloane decided to go online –first as an <a href="/wiki/Email" title="Email">email</a> service (August 1994), and soon thereafter as a website (1996). As a spin-off from the database work, Sloane founded the <a href="/wiki/Journal_of_Integer_Sequences" title="Journal of Integer Sequences">Journal of Integer Sequences</a> in 1998.<a href="#cite_note-10">[10]</a> The database continues to grow at a rate of some 10,000 entries a year. Sloane has personally managed 'his' sequences for almost 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the omnibus database.<a href="#cite_note-11">[11]</a> In 2004, Sloane celebrated the addition of the 100,000th sequence to the database, <a href="//oeis.org/A100000" class="extiw" title="oeis:A100000">A100000</a>, which counts the marks on the <a href="/wiki/Ishango_bone" title="Ishango bone">Ishango bone</a>. In 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an <a rel="nofollow" class="external text" href="//oeis.org/wiki/">OEIS wiki</a> at <a rel="nofollow" class="external text" href="//oeis.org/">OEIS.org</a> was created to simplify the collaboration of the OEIS editors and contributors.<a href="#cite_note-12">[12]</a> The 200,000th sequence, <a href="//oeis.org/A200000" class="extiw" title="oeis:A200000">A200000</a>, was added to the database in November 2011; it was initially entered as A200715, and moved to A200000 after a week of discussion on the SeqFan mailing list,<a href="#cite_note-13">[13]</a><a href="#cite_note-14">[14]</a> following a proposal by OEIS Editor-in-Chief <a href="/w/index.php?title=Charles_Greathouse&action=edit&redlink=1" class="new" title="Charles Greathouse (page does not exist)">Charles Greathouse</a> to choose a special sequence for A200000.<a href="#cite_note-15">[15]</a> A300000 was defined in February 2018, and by end of January 2023 the database contained more than 360,000 sequences.<a href="#cite_note-MATH_VALUES_2023_Sloane-16">[16]</a><a href="#cite_note-Sloane_2023_pp._193%E2%80%93205-17">[17]</a> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"><h2 id="Non-integers">Non-integers</h2> <a role="button" href="/w/index.php?title=On-Line_Encyclopedia_of_Integer_Sequences&action=edit&section=2" title="Edit section: Non-integers" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-2 collapsible-block" id="mf-section-2"> Besides integer sequences, the OEIS also catalogs sequences of <a href="/wiki/Fraction" title="Fraction">fractions</a>, the digits of <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental numbers</a>, <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> and so on by transforming them into integer sequences. Sequences of fractions are represented by two sequences (named with the keyword 'frac'): the sequence of numerators and the sequence of denominators. For example, the fifth-order <a href="/wiki/Farey_sequence" title="Farey sequence">Farey sequence</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {1 \over 5},{1 \over 4},{1 \over 3},{2 \over 5},{1 \over 2},{3 \over 5},{2 \over 3},{3 \over 4},{4 \over 5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {1 \over 5},{1 \over 4},{1 \over 3},{2 \over 5},{1 \over 2},{3 \over 5},{2 \over 3},{3 \over 4},{4 \over 5}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55b4378a031268d88057ef5101972d85c9497d48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:23.195ex; height:3.676ex;" alt="{\displaystyle \textstyle {1 \over 5},{1 \over 4},{1 \over 3},{2 \over 5},{1 \over 2},{3 \over 5},{2 \over 3},{3 \over 4},{4 \over 5}}"></noscript><span class="lazy-image-placeholder" style="width: 23.195ex;height: 3.676ex;vertical-align: -1.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55b4378a031268d88057ef5101972d85c9497d48" data-alt="{\displaystyle \textstyle {1 \over 5},{1 \over 4},{1 \over 3},{2 \over 5},{1 \over 2},{3 \over 5},{2 \over 3},{3 \over 4},{4 \over 5}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , is catalogued as the numerator sequence 1, 1, 1, 2, 1, 3, 2, 3, 4 (<a href="//oeis.org/A006842" class="extiw" title="oeis:A006842">A006842</a>) and the denominator sequence 5, 4, 3, 5, 2, 5, 3, 4, 5 (<a href="//oeis.org/A006843" class="extiw" title="oeis:A006843">A006843</a>). Important <a href="/wiki/Irrational_number" title="Irrational number">irrational numbers</a> such as π = 3.1415926535897... are catalogued under representative integer sequences such as <a href="/wiki/Decimal" title="Decimal">decimal</a> expansions (here 3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 3, 8, 4, 6, 2, 6, 4, 3, 3, 8, 3, 2, 7, 9, 5, 0, 2, 8, 8, ... (<a href="//oeis.org/A000796" class="extiw" title="oeis:A000796">A000796</a>)), <a href="/wiki/Binary_number" title="Binary number">binary</a> expansions (here 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, ... (<a href="//oeis.org/A004601" class="extiw" title="oeis:A004601">A004601</a>)), or <a href="/wiki/Simple_continued_fraction" title="Simple continued fraction">continued fraction expansions</a> (here 3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, ... (<a href="//oeis.org/A001203" class="extiw" title="oeis:A001203">A001203</a>)). </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"><h2 id="Conventions">Conventions</h2> <a role="button" href="/w/index.php?title=On-Line_Encyclopedia_of_Integer_Sequences&action=edit&section=3" title="Edit section: Conventions" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-3 collapsible-block" id="mf-section-3"> The OEIS was limited to plain <a href="/wiki/ASCII" title="ASCII">ASCII</a> text until 2011, and it still uses a linear form of conventional mathematical notation (such as f(n) for <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a>, n for running <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a>, etc.). <a href="/wiki/Greek_alphabet" title="Greek alphabet">Greek letters</a> are usually represented by their full names, e.g., mu for μ, phi for φ. Every sequence is identified by the letter A followed by six digits, almost always referred to with leading zeros, e.g., A000315 rather than A315. Individual terms of sequences are separated by commas. Digit groups are not separated by commas, periods, or spaces. In comments, formulas, etc., <code>a(n)</code> represents the nth term of the sequence. <div class="mw-heading mw-heading3"><h3 id="Special_meaning_of_zero">Special meaning of zero</h3> <a role="button" href="/w/index.php?title=On-Line_Encyclopedia_of_Integer_Sequences&action=edit&section=4" title="Edit section: Special meaning of zero" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <a href="/wiki/Zero" class="mw-redirect" title="Zero">Zero</a> is often used to represent non-existent sequence elements. For example, <a href="//oeis.org/A104157" class="extiw" title="oeis:A104157">A104157</a> enumerates the "smallest <a href="/wiki/Prime_number" title="Prime number">prime</a> of n2 consecutive primes to form an n × n <a href="/wiki/Magic_square" title="Magic square">magic square</a> of least <a href="/wiki/Magic_constant" title="Magic constant">magic constant</a>, or 0 if no such magic square exists." The value of a(1) (a 1 × 1 magic square) is 2; a(3) is 1480028129. But there is no such 2 × 2 magic square, so a(2) is 0. This special usage has a solid mathematical basis in certain counting functions; for example, the <a href="/wiki/Totient" class="mw-redirect" title="Totient">totient</a> valence function Nφ(m) (<a href="//oeis.org/A014197" class="extiw" title="oeis:A014197">A014197</a>) counts the solutions of φ(x) = m. There are 4 solutions for 4, but no solutions for 14, hence a(14) of A014197 is 0—there are no solutions. Other values are also used, most commonly −1 (see <a href="//oeis.org/A000230" class="extiw" title="oeis:A000230">A000230</a> or <a href="//oeis.org/A094076" class="extiw" title="oeis:A094076">A094076</a>). <div class="mw-heading mw-heading3"><h3 id="Lexicographical_ordering">Lexicographical ordering</h3> <a role="button" href="/w/index.php?title=On-Line_Encyclopedia_of_Integer_Sequences&action=edit&section=5" title="Edit section: Lexicographical ordering" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> The OEIS maintains the <a href="/wiki/Lexicographical_order" class="mw-redirect" title="Lexicographical order">lexicographical order</a> of the sequences, so each sequence has a predecessor and a successor (its "context").<a href="#cite_note-18">[18]</a> OEIS normalizes the sequences for lexicographical ordering, (usually) ignoring all initial zeros and ones, and also the <a href="/wiki/Sign_(mathematics)" title="Sign (mathematics)">sign</a> of each element. Sequences of <a href="/wiki/Weight_distribution" title="Weight distribution">weight distribution</a> codes often omit periodically recurring zeros. For example, consider: the <a href="/wiki/Prime_number" title="Prime number">prime numbers</a>, the <a href="/wiki/Palindromic_prime" title="Palindromic prime">palindromic primes</a>, the <a href="/wiki/Fibonacci_number" class="mw-redirect" title="Fibonacci number">Fibonacci sequence</a>, the <a href="/wiki/Lazy_caterer%27s_sequence" title="Lazy caterer's sequence">lazy caterer's sequence</a>, and the coefficients in the <a href="/wiki/Series_expansion" title="Series expansion">series expansion</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {{\zeta (n+2)} \over {\zeta (n)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {{\zeta (n+2)} \over {\zeta (n)}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91105db3ecd3c5953f16d8a0a5176c3bc088770b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.977ex; height:4.843ex;" alt="{\displaystyle \textstyle {{\zeta (n+2)} \over {\zeta (n)}}}"></noscript><span class="lazy-image-placeholder" style="width: 5.977ex;height: 4.843ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91105db3ecd3c5953f16d8a0a5176c3bc088770b" data-alt="{\displaystyle \textstyle {{\zeta (n+2)} \over {\zeta (n)}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . In OEIS lexicographic order, they are: <ul><li>Sequence #1: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, ... <a href="//oeis.org/A000040" class="extiw" title="oeis:A000040">A000040</a></li> <li>Sequence #2: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, ... <a href="//oeis.org/A002385" class="extiw" title="oeis:A002385">A002385</a></li> <li>Sequence #3: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, ... <a href="//oeis.org/A000045" class="extiw" title="oeis:A000045">A000045</a></li> <li>Sequence #4: 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, ... <a href="//oeis.org/A000124" class="extiw" title="oeis:A000124">A000124</a></li> <li>Sequence #5: 1, −3, −8, −3, −24, 24, −48, −3, −8, 72, −120, 24, −168, 144, ... <a href="//oeis.org/A046970" class="extiw" title="oeis:A046970">A046970</a></li></ul> whereas unnormalized lexicographic ordering would order these sequences thus: #3, #5, #4, #1, #2. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"><h2 id="Self-referential_sequences">Self-referential sequences</h2> <a role="button" href="/w/index.php?title=On-Line_Encyclopedia_of_Integer_Sequences&action=edit&section=6" title="Edit section: Self-referential sequences" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-4 collapsible-block" id="mf-section-4"> Very early in the history of the OEIS, sequences defined in terms of the numbering of sequences in the OEIS itself were proposed. "I resisted adding these sequences for a long time, partly out of a desire to maintain the dignity of the database, and partly because A22 was only known to 11 terms!", Sloane reminisced.<a href="#cite_note-19">[19]</a> One of the earliest self-referential sequences Sloane accepted into the OEIS was <a href="//oeis.org/A031135" class="extiw" title="oeis:A031135">A031135</a> (later <a href="//oeis.org/A091967" class="extiw" title="oeis:A091967">A091967</a>) "a(n) = n-th term of sequence An or –1 if An has fewer than n terms". This sequence spurred progress on finding more terms of <a href="//oeis.org/A000022" class="extiw" title="oeis:A000022">A000022</a>. <a href="//oeis.org/A100544" class="extiw" title="oeis:A100544">A100544</a> lists the first term given in sequence An, but it needs to be updated from time to time because of changing opinions on offsets. Listing instead term a(1) of sequence An might seem a good alternative if it were not for the fact that some sequences have offsets of 2 and greater. This line of thought leads to the question "Does sequence An contain the number n?" and the sequences <a href="//oeis.org/A053873" class="extiw" title="oeis:A053873">A053873</a>, "Numbers n such that OEIS sequence An contains n", and <a href="//oeis.org/A053169" class="extiw" title="oeis:A053169">A053169</a>, "n is in this sequence <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> n is not in sequence An". Thus, the <a href="/wiki/Composite_number" title="Composite number">composite number</a> 2808 is in A053873 because <a href="//oeis.org/A002808" class="extiw" title="oeis:A002808">A002808</a> is the sequence of composite numbers, while the non-prime 40 is in A053169 because it is not in <a href="//oeis.org/A000040" class="extiw" title="oeis:A000040">A000040</a>, the prime numbers. Each n is a member of exactly one of these two sequences, and in principle it can be determined which sequence each n belongs to, with two exceptions (related to the two sequences themselves): <ul><li>It cannot be determined whether 53873 is a member of A053873 or not. If it is in the sequence then by definition it should be; if it is not in the sequence then (again, by definition) it should not be. Nevertheless, either decision would be consistent, and would also resolve the question of whether 53873 is in A053169.</li> <li>It can be proved that 53169 <a href="/wiki/Principle_of_contradiction" class="mw-redirect" title="Principle of contradiction">both is and is not</a> a member of A053169. If it is in the sequence then by definition it should not be; if it is not in the sequence then (again, by definition) it should be. This is a form of <a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a>. Hence it is also not possible to answer if 53169 is in A053873.</li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"><h2 id="Abridged_example_of_a_comprehensive_entry">Abridged example of a comprehensive entry</h2> <a role="button" href="/w/index.php?title=On-Line_Encyclopedia_of_Integer_Sequences&action=edit&section=7" title="Edit section: Abridged example of a comprehensive entry" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-5 collapsible-block" id="mf-section-5"> This entry, <a href="//oeis.org/A046970" class="extiw" title="oeis:A046970">A046970</a>, was chosen because it comprehensively contains every OEIS field, filled. <a href="#cite_note-20">[20]</a> <div style="overflow: auto;" class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre>A046970 Dirichlet inverse of the Jordan function J_2 (A007434). 1, -3, -8, -3, -24, 24, -48, -3, -8, 72, -120, 24, -168, 144, 192, -3, -288, 24, -360, 72, 384, 360, -528, 24, -24, 504, -8, 144, -840, -576, -960, -3, 960, 864, 1152, 24, -1368, 1080, 1344, 72, -1680, -1152, -1848, 360, 192, 1584, -2208, 24, -48, 72, 2304, 504, -2808, 24, 2880, 144, 2880, 2520, -3480, -576 OFFSET 1,2 COMMENTS B(n+2) = -B(n)*((n+2)*(n+1)/(4*Pi^2))*z(n+2)/z(n) = -B(n)*((n+2)*(n+1)/(4*Pi^2)) * Sum_{j>=1} a(j)/j^(n+2). Apart from signs also Sum_{d|n} core(d)^2*mu(n/d) where core(x) is the squarefree part of x. - Benoit Cloitre, May 31 2002 REFERENCES M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, 1965, pp. 805-811. T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986, p. 48. LINKS Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. P. G. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 (516) (2005) 403-408. Paul W. Oxby, A Function Based on Chebyshev Polynomials as an Alternative to the Sinc Function in FIR Filter Design, arXiv:2011.10546 [eess.SP], 2020. Wikipedia, Riemann zeta function. FORMULA Multiplicative with a(p^e) = 1 - p^2. a(n) = Sum_{d|n} mu(d)*d^2. abs(a(n)) = Product_{p prime divides n} (p^2 - 1). - Jon Perry, Aug 24 2010 From Wolfdieter Lang, Jun 16 2011: (Start) Dirichlet g.f.: zeta(s)/zeta(s-2). a(n) = J_{-2}(n)*n^2, with the Jordan function J_k(n), with J_k(1):=1. See the Apostol reference, p. 48. exercise 17. (End) a(prime(n)) = -A084920(n). - R. J. Mathar, Aug 28 2011 G.f.: Sum_{k>=1} mu(k)*k^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 15 2017 EXAMPLE a(3) = -8 because the divisors of 3 are {1, 3} and mu(1)*1^2 + mu(3)*3^2 = -8. a(4) = -3 because the divisors of 4 are {1, 2, 4} and mu(1)*1^2 + mu(2)*2^2 + mu(4)*4^2 = -3. E.g., a(15) = (3^2 - 1) * (5^2 - 1) = 8*24 = 192. - Jon Perry, Aug 24 2010 G.f. = x - 3*x^2 - 8*x^3 - 3*x^4 - 24*x^5 + 24*x^6 - 48*x^7 - 3*x^8 - 8*x^9 + ... MAPLE Jinvk := proc(n, k) local a, f, p ; a := 1 ; for f in ifactors(n)[2] do p := op(1, f) ; a := a*(1-p^k) ; end do: a ; end proc: A046970 := proc(n) Jinvk(n, 2) ; end proc: # R. J. Mathar, Jul 04 2011 MATHEMATICA muDD[d_] := MoebiusMu[d]*d^2; Table[Plus @@ muDD[Divisors[n]], {n, 60}] (Lopez) Flatten[Table[{ x = FactorInteger[n]; p = 1; For[i = 1, i <= Length[x], i++, p = p*(1 - x[[i]][[1]]^2)]; p}, {n, 1, 50, 1}]] (* Jon Perry, Aug 24 2010 *) a[ n_] := If[ n < 1, 0, Sum[ d^2 MoebiusMu[ d], {d, Divisors @ n}]] (* Michael Somos, Jan 11 2014 *) a[ n_] := If[ n < 2, Boole[ n == 1], Times @@ (1 - #[[1]]^2 & /@ FactorInteger @ n)] (* Michael Somos, Jan 11 2014 *) PROG (PARI) A046970(n)=sumdiv(n, d, d^2*moebius(d)) \\ Benoit Cloitre (Haskell) a046970 = product . map ((1 -) . (^ 2)) . a027748_row -- Reinhard Zumkeller, Jan 19 2012 (PARI) {a(n) = if( n<1, 0, direuler( p=2, n, (1 - X*p^2) / (1 - X))[n])} /* Michael Somos, Jan 11 2014 */ CROSSREFS Cf. A007434, A027641, A027642, A063453, A023900. Cf. A027748. Sequence in context: A144457 A220138 A146975 * A322360 A058936 A280369 Adjacent sequences: A046967 A046968 A046969 * A046971 A046972 A046973 KEYWORD sign,easy,mult AUTHOR Douglas Stoll, dougstoll(AT)email.msn.com EXTENSIONS Corrected and extended by Vladeta Jovovic, Jul 25 2001 Additional comments from Wilfredo Lopez (chakotay147138274(AT)yahoo.com), Jul 01 2005 </pre></div> <div class="mw-heading mw-heading3"><h3 id="Entry_fields">Entry fields</h3> <a role="button" href="/w/index.php?title=On-Line_Encyclopedia_of_Integer_Sequences&action=edit&section=8" title="Edit section: Entry fields" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <dl><dt>ID number</dt> <dd>Every sequence in the OEIS has a <a href="/wiki/Serial_number" title="Serial number">serial number</a>, a six-digit positive <a href="/wiki/Integer" title="Integer">integer</a>, prefixed by A (and zero-padded on the left prior to November 2004). The letter "A" stands for "absolute". Numbers are either assigned by the editor(s) or by an A number dispenser, which is handy for when contributors wish to send in multiple related sequences at once and be able to create cross-references. An A number from the dispenser expires a month from issue if not used. But as the following table of arbitrarily selected sequences shows, the rough correspondence holds.</dd></dl> <table class="wikitable" align="center"> <tbody><tr> <th><a href="//oeis.org/A059097" class="extiw" title="oeis:A059097">A059097</a> </th> <td>Numbers n such that the <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficient</a> C(2n, n) is not <a href="/wiki/Divisible" class="mw-redirect" title="Divisible">divisible</a> by the <a href="/wiki/Square_(algebra)" title="Square (algebra)">square</a> of an <a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">odd</a> prime. </td> <td>Jan 1, 2001 </td></tr> <tr> <th><a href="//oeis.org/A060001" class="extiw" title="oeis:A060001">A060001</a> </th> <td><a href="/wiki/Fibonacci_number" class="mw-redirect" title="Fibonacci number">Fibonacci</a>(n)!. </td> <td>Mar 14, 2001 </td></tr> <tr> <th><a href="//oeis.org/A066288" class="extiw" title="oeis:A066288">A066288</a> </th> <td>Number of 3-dimensional <a href="/wiki/Polyomino" title="Polyomino">polyominoes</a> (or <a href="/wiki/Polycube" title="Polycube">polycubes</a>) with n cells and symmetry group of <a href="/wiki/Order_(group_theory)" title="Order (group theory)">order</a> exactly 24. </td> <td>Jan 1, 2002 </td></tr> <tr> <th><a href="//oeis.org/A075000" class="extiw" title="oeis:A075000">A075000</a> </th> <td>Smallest number such that n · a(n) is a concatenation of n consecutive integers ... </td> <td>Aug 31, 2002 </td></tr> <tr> <th><a href="//oeis.org/A078470" class="extiw" title="oeis:A078470">A078470</a> </th> <td>Continued fraction for ζ(3/2) </td> <td>Jan 1, 2003 </td></tr> <tr> <th><a href="//oeis.org/A080000" class="extiw" title="oeis:A080000">A080000</a> </th> <td>Number of permutations satisfying −k ≤ p(i) − i ≤ r and p(i) − i </td> <td>Feb 10, 2003 </td></tr> <tr> <th><a href="//oeis.org/A090000" class="extiw" title="oeis:A090000">A090000</a> </th> <td>Length of longest contiguous block of 1s in binary expansion of nth prime. </td> <td>Nov 20, 2003 </td></tr> <tr> <th><a href="//oeis.org/A091345" class="extiw" title="oeis:A091345">A091345</a> </th> <td>Exponential convolution of A069321(n) with itself, where we set A069321(0) = 0. </td> <td>Jan 1, 2004 </td></tr> <tr> <th><a href="//oeis.org/A100000" class="extiw" title="oeis:A100000">A100000</a> </th> <td>Marks from the 22000-year-old <a href="/wiki/Ishango_bone" title="Ishango bone">Ishango bone</a> from the Congo. </td> <td>Nov 7, 2004 </td></tr> <tr> <th><a href="//oeis.org/A102231" class="extiw" title="oeis:A102231">A102231</a> </th> <td>Column 1 of triangle A102230, and equals the convolution of A032349 with A032349 shift right. </td> <td>Jan 1, 2005 </td></tr> <tr> <th><a href="//oeis.org/A110030" class="extiw" title="oeis:A110030">A110030</a> </th> <td>Number of consecutive integers starting with n needed to sum to a Niven number. </td> <td>Jul 8, 2005 </td></tr> <tr> <th><a href="//oeis.org/A112886" class="extiw" title="oeis:A112886">A112886</a> </th> <td>Triangle-free positive integers. </td> <td>Jan 12, 2006 </td></tr> <tr> <th><a href="//oeis.org/A120007" class="extiw" title="oeis:A120007">A120007</a> </th> <td><a href="/wiki/M%C3%B6bius_transform" class="mw-redirect" title="Möbius transform">Möbius transform</a> of sum of prime <a href="/wiki/Divisor" title="Divisor">factors</a> of n with multiplicity. </td> <td>Jun 2, 2006 </td></tr></tbody></table> <dl><dd>Even for sequences in the book predecessors to the OEIS, the ID numbers are not the same. The 1973 Handbook of Integer Sequences contained about 2400 sequences, which were numbered by lexicographic order (the letter N plus four digits, zero-padded where necessary), and the 1995 Encyclopedia of Integer Sequences contained 5487 sequences, also numbered by lexicographic order (the letter M plus 4 digits, zero-padded where necessary). These old M and N numbers, as applicable, are contained in the ID number field in parentheses after the modern A number.</dd> <dt>Sequence data</dt> <dd>The sequence field lists the numbers themselves, to about 260 characters.<a href="#cite_note-21">[21]</a> More terms of the sequences can be provided in so-called B-files.<a href="#cite_note-22">[22]</a> The sequence field makes no distinction between sequences that are finite but still too long to display and sequences that are infinite; instead, the keywords "fini", "full", and "more" are used to distinguish such sequences. To determine to which n the values given correspond, see the offset field, which gives the n for the first term given.</dd> <dt>Name</dt> <dd>The name field usually contains the most common name for the sequence, and sometimes also the formula. For example, 1, 8, 27, 64, 125, 216, 343, 512, (<a href="//oeis.org/A000578" class="extiw" title="oeis:A000578">A000578</a>) is named "The <a href="/wiki/Cube_(algebra)" title="Cube (algebra)">cubes</a>: a(n) = n^3.".</dd> <dt>Comments</dt> <dd>The comments field is for information about the sequence that does not quite fit in any of the other fields. The comments field often points out interesting relationships between different sequences and less obvious applications for a sequence. For example, Lekraj Beedassy in a comment to A000578 notes that the cube numbers also count the "total number of <a href="/wiki/Triangle" title="Triangle">triangles</a> resulting from criss-crossing <a href="/wiki/Cevian" title="Cevian">cevians</a> within a triangle so that two of its sides are each n-partitioned", and Neil Sloane points out an unexpected relationship between <a href="/wiki/Centered_hexagonal_number" title="Centered hexagonal number">centered hexagonal numbers</a> (<a href="//oeis.org/A003215" class="extiw" title="oeis:A003215">A003215</a>) and second <a href="/wiki/Bessel_polynomials" title="Bessel polynomials">Bessel polynomials</a> (<a href="//oeis.org/A001498" class="extiw" title="oeis:A001498">A001498</a>) in a comment to A003215.</dd> <dt>References</dt> <dd>References to printed documents (books, papers, ...).</dd> <dt>Links</dt> <dd>Links, i.e. <a href="/wiki/Uniform_Resource_Locator" class="mw-redirect" title="Uniform Resource Locator">URLs</a>, to online resources. These may be: <ol><li>references to applicable articles in journals</li> <li>links to the index</li> <li>links to text files which hold the sequence terms (in a two column format) over a wider range of indices than held by the main database lines</li> <li>links to images in the local database directories which often provide combinatorial background related to <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a></li> <li>others related to computer codes, more extensive tabulations in specific research areas provided by individuals or research groups</li></ol></dd></dl> <dl><dt>Formula</dt> <dd>Formulas, <a href="/wiki/Recurrence_relation" title="Recurrence relation">recurrences</a>, <a href="/wiki/Generating_function" title="Generating function">generating functions</a>, etc. for the sequence.</dd> <dt>Example</dt> <dd>Some examples of sequence member values.</dd> <dt>Maple</dt> <dd><a href="/wiki/Maple_computer_algebra_system" class="mw-redirect" title="Maple computer algebra system">Maple</a> code.</dd> <dt>Mathematica</dt> <dd><a href="/wiki/Wolfram_Language" title="Wolfram Language">Wolfram Language</a> code.</dd> <dt>Program</dt> <dd>Originally <a href="/wiki/Maple_computer_algebra_system" class="mw-redirect" title="Maple computer algebra system">Maple</a> and <a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a> were the preferred programs for calculating sequences in the OEIS, each with their own field labels. As of 2016<a class="external text" href="https://en.wikipedia.org/w/index.php?title=On-Line_Encyclopedia_of_Integer_Sequences&action=edit">[update]</a>, Mathematica was the most popular choice with 100,000 Mathematica programs followed by 50,000 <a href="/wiki/PARI/GP" title="PARI/GP">PARI/GP</a> programs, 35,000 Maple programs, and 45,000 in other languages.</dd> <dd>As for any other part of the record, if there is no name given, the contribution (here: program) was written by the original submitter of the sequence.</dd> <dt>Crossrefs</dt> <dd>Sequence cross-references originated by the original submitter are usually denoted by "<a href="/wiki/Cf." title="Cf.">Cf.</a>"</dd> <dd>Except for new sequences, the "see also" field also includes information on the lexicographic order of the sequence (its "context") and provides links to sequences with close A numbers (A046967, A046968, A046969, A046971, A046972, A046973, in our example). The following table shows the context of our example sequence, A046970:</dd></dl> <table class="wikitable" align="center"> <tbody><tr> <th><a href="//oeis.org/A016623" class="extiw" title="oeis:A016623">A016623</a> </th> <td>3, 8, 3, 9, 4, 5, 2, 3, 1, 2, ... </td> <td>Decimal expansion of <a href="/wiki/Natural_logarithm" title="Natural logarithm">ln</a>(93/2). </td></tr> <tr> <th><a href="//oeis.org/A046543" class="extiw" title="oeis:A046543">A046543</a> </th> <td>1, 1, 1, 3, 8, 3, 10, 1, 110, 3, 406, 3 </td> <td>First numerator and then denominator of the central elements of the 1/3-Pascal triangle (by row). </td></tr> <tr> <th><a href="//oeis.org/A035292" class="extiw" title="oeis:A035292">A035292</a> </th> <td>1, 3, 8, 3, 12, 24, 16, 3, 41, 36, 24, ... </td> <td>Number of similar sublattices of Z4 of index n2. </td></tr> <tr> <th><a href="//oeis.org/A046970" class="extiw" title="oeis:A046970">A046970</a> </th> <td>1, −3, −8, −3, −24, 24, −48, −3, −8, 72, ... </td> <td>Generated from <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>... </td></tr> <tr> <th><a href="//oeis.org/A058936" class="extiw" title="oeis:A058936">A058936</a> </th> <td>0, 1, 3, 8, 3, 30, 20, 144, 90, 40, 840, 504, 420, 5760, 3360, 2688, 1260 </td> <td>Decomposition of Stirling's S(n, 2) based on associated numeric partitions. </td></tr> <tr> <th><a href="//oeis.org/A002017" class="extiw" title="oeis:A002017">A002017</a> </th> <td>1, 1, 1, 0, −3, −8, −3, 56, 217, 64, −2951, −12672, ... </td> <td>Expansion of <a href="/wiki/Exponential_function" title="Exponential function">exp</a>(<a href="/wiki/Sine" class="mw-redirect" title="Sine">sin</a> x). </td></tr> <tr> <th><a href="//oeis.org/A086179" class="extiw" title="oeis:A086179">A086179</a> </th> <td>3, 8, 4, 1, 4, 9, 9, 0, 0, 7, 5, 4, 3, 5, 0, 7, 8 </td> <td>Decimal expansion of upper bound for the r-values supporting stable period-3 orbits in the <a href="/wiki/Logistic_map" title="Logistic map">logistic map</a>. </td></tr></tbody></table> <dl><dt>Keyword</dt> <dd>The OEIS has its own <a href="/wiki/Lexicon" title="Lexicon">lexicon</a>: a standard set of mostly four-letter keywords which <a href="/wiki/Taxonomy" title="Taxonomy">characterizes</a> each sequence:<a href="#cite_note-terms-explanation-23">[23]</a> <ul><li>allocated - An A-number which has been set aside for a user but for which the entry has not yet been approved (and perhaps not yet written).</li> <li>base - The results of the calculation depend on a specific <a href="/wiki/Positional_notation" title="Positional notation">positional base</a>. For example, 2, 3, 5, 7, 11, 101, 131, 151, 181 ... <a href="//oeis.org/A002385" class="extiw" title="oeis:A002385">A002385</a> are prime numbers regardless of base, but they are <a href="/wiki/Palindromic_prime" title="Palindromic prime">palindromic</a> specifically in base 10. Most of them are not palindromic in binary. Some sequences rate this keyword depending on how they are defined. For example, the <a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne primes</a> 3, 7, 31, 127, 8191, 131071, ... <a href="//oeis.org/A000668" class="extiw" title="oeis:A000668">A000668</a> does not rate "base" if defined as "primes of the form 2^n − 1". However, defined as "<a href="/wiki/Repunit" title="Repunit">repunit</a> primes in binary," the sequence would rate the keyword "base".</li> <li>bref - "sequence is too short to do any analysis with", for example, <a href="//oeis.org/A079243" class="extiw" title="oeis:A079243">A079243</a>, the number of <a href="/wiki/Isomorphism_class" class="mw-redirect" title="Isomorphism class">isomorphism classes</a> of <a href="/wiki/Associative" class="mw-redirect" title="Associative">associative</a> non-<a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a> non-anti-associative <a href="/wiki/Anti-commutative" class="mw-redirect" title="Anti-commutative">anti-commutative</a> closed <a href="/wiki/Binary_operation" title="Binary operation">binary operations</a> on a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of order n.</li> <li>changed The sequence is changed in the last two weeks.</li> <li>cofr - The sequence represents a <a href="/wiki/Continued_fraction" title="Continued fraction">continued fraction</a>, for example the continued fraction expansion of e (<a href="//oeis.org/A003417" class="extiw" title="oeis:A003417">A003417</a>) or π (<a href="//oeis.org/A001203" class="extiw" title="oeis:A001203">A001203</a>).</li> <li>cons - The sequence is a decimal expansion of a <a href="/wiki/Mathematical_constant" title="Mathematical constant">mathematical constant</a>, such as e (<a href="//oeis.org/A001113" class="extiw" title="oeis:A001113">A001113</a>) or π (<a href="//oeis.org/A000796" class="extiw" title="oeis:A000796">A000796</a>).</li> <li>core - A sequence that is of foundational importance to a branch of mathematics, such as the prime numbers (<a href="//oeis.org/A000040" class="extiw" title="oeis:A000040">A000040</a>), the Fibonacci sequence (<a href="//oeis.org/A000045" class="extiw" title="oeis:A000045">A000045</a>), etc.</li> <li>dead - This keyword used for erroneous sequences that have appeared in papers or books, or for duplicates of existing sequences. For example, <a href="//oeis.org/A088552" class="extiw" title="oeis:A088552">A088552</a> is the same as <a href="//oeis.org/A000668" class="extiw" title="oeis:A000668">A000668</a>.</li> <li>dumb - One of the more subjective keywords, for "unimportant sequences," which may or may not directly relate to mathematics, such as <a href="/wiki/Popular_culture" title="Popular culture">popular culture</a> references, arbitrary sequences from Internet puzzles, and sequences related to <a href="/wiki/Numeric_keypad" title="Numeric keypad">numeric keypad</a> entries. <a href="//oeis.org/A001355" class="extiw" title="oeis:A001355">A001355</a>, "Mix digits of pi and e" is one example of lack of importance, and <a href="//oeis.org/A085808" class="extiw" title="oeis:A085808">A085808</a>, "Price is Right wheel" (the sequence of numbers on the <a href="/wiki/Showcase_Showdown" class="mw-redirect" title="Showcase Showdown">Showcase Showdown</a> wheel used in the U.S. game show <a href="/wiki/The_Price_Is_Right_(U.S._game_show)" class="mw-redirect" title="The Price Is Right (U.S. game show)">The Price Is Right</a>) is an example of a non-mathematics-related sequence, kept mainly for trivia purposes.<a href="#cite_note-24">[24]</a></li> <li>easy - The terms of the sequence can be easily calculated. Perhaps the sequence most deserving of this keyword is 1, 2, 3, 4, 5, 6, 7, ... <a href="//oeis.org/A000027" class="extiw" title="oeis:A000027">A000027</a>, where each term is 1 more than the previous term. The keyword "easy" is sometimes given to sequences "primes of the form f(m)" where f(m) is an easily calculated function. (Though even if f(m) is easy to calculate for large m, it might be very difficult to determine if f(m) is prime).</li> <li>eigen - A sequence of <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a>.</li> <li>fini - The sequence is finite, although it might still contain more terms than can be displayed. For example, the sequence field of <a href="//oeis.org/A105417" class="extiw" title="oeis:A105417">A105417</a> shows only about a quarter of all the terms, but a comment notes that the last term is 3888.</li> <li>frac - A sequence of either numerators or denominators of a sequence of fractions representing <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>. Any sequence with this keyword ought to be cross-referenced to its matching sequence of numerators or denominators, though this may be dispensed with for sequences of <a href="/wiki/Egyptian_fraction" title="Egyptian fraction">Egyptian fractions</a>, such as <a href="//oeis.org/A069257" class="extiw" title="oeis:A069257">A069257</a>, where the sequence of numerators would be <a href="//oeis.org/A000012" class="extiw" title="oeis:A000012">A000012</a>. This keyword should not be used for sequences of continued fractions; cofr should be used instead for that purpose.</li> <li>full - The sequence field displays the complete sequence. If a sequence has the keyword "full", it should also have the keyword "fini". One example of a finite sequence given in full is that of the <a href="/wiki/Supersingular_prime_(moonshine_theory)" title="Supersingular prime (moonshine theory)">supersingular primes</a> <a href="//oeis.org/A002267" class="extiw" title="oeis:A002267">A002267</a>, of which there are precisely fifteen.</li> <li>hard - The terms of the sequence cannot be easily calculated, even with raw number crunching power. This keyword is most often used for sequences corresponding to unsolved problems, such as "How many <a href="/wiki/N-sphere" title="N-sphere">n-spheres</a> can touch another n-sphere of the same size?" <a href="//oeis.org/A001116" class="extiw" title="oeis:A001116">A001116</a> lists the first ten known solutions.</li> <li>hear - A sequence with a graph audio deemed to be "particularly interesting and/or beautiful", some examples are collected at the <a rel="nofollow" class="external text" href="https://oeis.org/play.html">OEIS site</a>.</li> <li>less - A "less interesting sequence".</li> <li>look - A sequence with a graph visual deemed to be "particularly interesting and/or beautiful". Two examples out of several thousands are <a rel="nofollow" class="external text" href="https://oeis.org/A331124/graph">A331124</a> <a rel="nofollow" class="external text" href="https://oeis.org/A347347/graph">A347347</a>.</li> <li>more - More terms of the sequence are wanted. Readers can submit an extension.</li> <li>mult - The sequence corresponds to a <a href="/wiki/Multiplicative_function" title="Multiplicative function">multiplicative function</a>. Term a(1) should be 1, and term a(mn) can be calculated by multiplying a(m) by a(n) if m and n are <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a>. For example, in <a href="//oeis.org/A046970" class="extiw" title="oeis:A046970">A046970</a>, a(12) = a(3)a(4) = −8 × −3.</li> <li>new - For sequences that were added in the last couple of weeks, or had a major extension recently. This keyword is not given a checkbox in the Web form for submitting new sequences; Sloane's program adds it by default where applicable.</li> <li>nice - Perhaps the most subjective keyword of all, for "<a href="/wiki/Mathematical_beauty" title="Mathematical beauty">exceptionally nice sequences</a>."</li> <li>nonn - The sequence consists of nonnegative integers (it may include zeroes). No distinction is made between sequences that consist of nonnegative numbers only because of the chosen offset (e.g., n3, the cubes, which are all nonnegative from n = 0 forwards) and those that by definition are completely nonnegative (e.g., n2, the squares).</li> <li>obsc - The sequence is considered obscure and needs a better definition.</li> <li>recycled - When the editors agree that a new proposed sequence is not worth adding to the OEIS, an editor blanks the entry leaving only the keyword line with keyword:recycled. The A-number then becomes available for allocation for another new sequence.</li> <li>sign - Some (or all) of the values of the sequence are negative. The entry includes both a Signed field with the signs and a Sequence field consisting of all the values passed through the <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> function.</li> <li>tabf - "An irregular (or funny-shaped) array of numbers made into a sequence by reading it row by row." For example, <a href="//oeis.org/A071031" class="extiw" title="oeis:A071031">A071031</a>, "Triangle read by rows giving successive states of <a href="/wiki/Cellular_automaton" title="Cellular automaton">cellular automaton</a> generated by "rule 62."</li> <li>tabl - A sequence obtained by reading a geometric arrangement of numbers, such as a triangle or square, row by row. The quintessential example is <a href="/wiki/Pascal%27s_triangle" title="Pascal's triangle">Pascal's triangle</a> read by rows, <a href="//oeis.org/A007318" class="extiw" title="oeis:A007318">A007318</a>.</li> <li>uned - The sequence has not been edited but it could be worth including in the OEIS. The sequence may contain computational or typographical errors. Contributors are encouraged to edit these sequences.</li> <li>unkn - "Little is known" about the sequence, not even the formula that produces it. For example, <a href="//oeis.org/A072036" class="extiw" title="oeis:A072036">A072036</a>, which was presented to the <a href="/wiki/Internet_Oracle" title="Internet Oracle">Internet Oracle</a> to ponder.</li> <li>walk - "Counts walks (or <a href="/wiki/Self-avoiding_walk" title="Self-avoiding walk">self-avoiding paths</a>)."</li> <li>word - Depends on the words of a specific language. For example, zero, one, two, three, four, five, etc. For example, 4, 3, 3, 5, 4, 4, 3, 5, 5, 4, 3, 6, 6, 8, 8, 7, 7, 9, 8, 8 ... <a href="//oeis.org/A005589" class="extiw" title="oeis:A005589">A005589</a>, "Number of letters in the English name of n, excluding spaces and hyphens."</li></ul></dd> <dd>Some keywords are mutually exclusive, namely: core and dumb, easy and hard, full and more, less and nice, and nonn and sign.</dd> <dt>Offset</dt> <dd>The offset is the index of the first term given. For some sequences, the offset is obvious. For example, if we list the sequence of square numbers as 0, 1, 4, 9, 16, 25 ..., the offset is 0; while if we list it as 1, 4, 9, 16, 25 ..., the offset is 1. The default offset is 0, and most sequences in the OEIS have offset of either 0 or 1. Sequence <a href="//oeis.org/A073502" class="extiw" title="oeis:A073502">A073502</a>, the <a href="/wiki/Magic_constant" title="Magic constant">magic constant</a> for n × n <a href="/wiki/Magic_square" title="Magic square">magic square</a> with prime entries (regarding 1 as a prime) with smallest row sums, is an example of a sequence with offset 3, and <a href="//oeis.org/A072171" class="extiw" title="oeis:A072171">A072171</a>, "Number of stars of visual magnitude n." is an example of a sequence with offset −1. Sometimes there can be disagreement over what the initial terms of the sequence are, and correspondingly what the offset should be. In the case of the <a href="/wiki/Lazy_caterer%27s_sequence" title="Lazy caterer's sequence">lazy caterer's sequence</a>, the maximum number of pieces you can cut a pancake into with n cuts, the OEIS gives the sequence as 1, 2, 4, 7, 11, 16, 22, 29, 37, ... <a href="//oeis.org/A000124" class="extiw" title="oeis:A000124">A000124</a>, with offset 0, while <a href="/wiki/Mathworld" class="mw-redirect" title="Mathworld">Mathworld</a> gives the sequence as 2, 4, 7, 11, 16, 22, 29, 37, ... (implied offset 1). It can be argued that making no cuts to the pancake is technically a number of cuts, namely n = 0, but it can also be argued that an uncut pancake is irrelevant to the problem. Although the offset is a required field, some contributors do not bother to check if the default offset of 0 is appropriate to the sequence they are sending in. The internal format actually shows two numbers for the offset. The first is the number described above, while the second represents the index of the first entry (counting from 1) that has an absolute value greater than 1. This second value is used to speed up the process of searching for a sequence. Thus <a href="//oeis.org/A000001" class="extiw" title="oeis:A000001">A000001</a>, which starts 1, 1, 1, 2 with the first entry representing a(1) has 1, 4 as the internal value of the offset field.</dd> <dt>Author(s)</dt> <dd>The author(s) of the sequence is (are) the person(s) who submitted the sequence, even if the sequence has been known since ancient times. The name of the submitter(s) is given first name (spelled out in full), middle initial(s) (if applicable) and last name; this in contrast to the way names are written in the reference fields. The e-mail address of the submitter is also given before 2011, with the @ character replaced by "(AT)" with some exceptions such as for associate editors or if an e-mail address does not exist. Now it has been the policy for OEIS not to display e-mail addresses in sequences. For most sequences after A055000, the author field also includes the date the submitter sent in the sequence.</dd> <dt>Extension</dt> <dd>Names of people who extended (added more terms to) the sequence or corrected terms of a sequence, followed by date of extension.</dd></dl> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"><h2 id="Sloane's_gap">Sloane's gap</h2> <a role="button" href="/w/index.php?title=On-Line_Encyclopedia_of_Integer_Sequences&action=edit&section=9" title="Edit section: Sloane's gap" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-6 collapsible-block" id="mf-section-6"> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sloanes_gap.png" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Sloanes_gap.png/220px-Sloanes_gap.png" decoding="async" width="220" height="145" class="mw-file-element" data-file-width="566" data-file-height="374"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 145px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Sloanes_gap.png/220px-Sloanes_gap.png" data-width="220" data-height="145" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Sloanes_gap.png/330px-Sloanes_gap.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Sloanes_gap.png/440px-Sloanes_gap.png 2x" data-class="mw-file-element"> </a><figcaption>Plot of Sloane's Gap: number of occurrences (y log scale) of each integer (x scale) in the OEIS database</figcaption></figure>In 2009, the OEIS database was used by Philippe Guglielmetti to measure the "importance" of each integer number.<a href="#cite_note-25">[25]</a> The result shown in the plot on the right shows a clear "gap" between two distinct point clouds,<a href="#cite_note-26">[26]</a> the "<a href="/wiki/Interesting_number_paradox" title="Interesting number paradox">uninteresting numbers</a>" (blue dots) and the "interesting" numbers that occur comparatively more often in sequences from the OEIS. It contains essentially prime numbers (red), numbers of the form an (green) and <a href="/wiki/Highly_composite_number" title="Highly composite number">highly composite numbers</a> (yellow). This phenomenon was studied by <a href="/w/index.php?title=Nicolas_Gauvrit&action=edit&redlink=1" class="new" title="Nicolas Gauvrit (page does not exist)">Nicolas Gauvrit</a>, <a href="/wiki/Jean-Paul_Delahaye" title="Jean-Paul Delahaye">Jean-Paul Delahaye</a> and Hector Zenil who explained the speed of the two clouds in terms of algorithmic complexity and the gap by social factors based on an artificial preference for sequences of primes, <a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">even</a> numbers, geometric and Fibonacci-type sequences and so on.<a href="#cite_note-27">[27]</a> Sloane's gap was featured on a <a href="/wiki/Numberphile" title="Numberphile">Numberphile</a> video in 2013.<a href="#cite_note-28">[28]</a> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"><h2 id="See_also">See also</h2> <a role="button" href="/w/index.php?title=On-Line_Encyclopedia_of_Integer_Sequences&action=edit&section=10" title="Edit section: See also" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-7 collapsible-block" id="mf-section-7"> <ul><li><a href="/wiki/List_of_OEIS_sequences" class="mw-redirect" title="List of OEIS sequences">List of OEIS sequences</a></li> <li><a href="/wiki/Abramowitz_and_Stegun" title="Abramowitz and Stegun">Abramowitz and Stegun</a></li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"><h2 id="Notes">Notes</h2> <a role="button" href="/w/index.php?title=On-Line_Encyclopedia_of_Integer_Sequences&action=edit&section=11" title="Edit section: Notes" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-8 collapsible-block" id="mf-section-8"> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-oeisfgoals-1"><a href="#cite_ref-oeisfgoals_1-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20131206172532/http://oeisf.org/index.html#GOALS">"Goals of The OEIS Foundation Inc"</a>. The OEIS Foundation Inc. Archived from <a rel="nofollow" class="external text" href="http://oeisf.org/index.html#GOALS">the original</a> on 2013-12-06. Retrieved 2017-11-06.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+OEIS+Foundation+Inc.&rft.atitle=Goals+of+The+OEIS+Foundation+Inc.&rft_id=http%3A%2F%2Foeisf.org%2Findex.html%23GOALS&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"> </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> Registration is required for editing entries or submitting new entries to the database </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://oeis.org/wiki/The_OEIS_End-User_License_Agreement">"The OEIS End-User License Agreement - OeisWiki"</a>. oeis.org. Retrieved 2023-02-26.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=oeis.org&rft.atitle=The+OEIS+End-User+License+Agreement+-+OeisWiki&rft_id=https%3A%2F%2Foeis.org%2Fwiki%2FThe_OEIS_End-User_License_Agreement&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"> </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20131206172532/http://oeisf.org/index.html#IPXFER">"Transfer of IP in OEIS to the OEIS Foundation Inc"</a>. Archived from <a rel="nofollow" class="external text" href="http://oeisf.org/index.html#IPXFER">the original</a> on 2013-12-06. Retrieved 2010-06-01.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Transfer+of+IP+in+OEIS+to+the+OEIS+Foundation+Inc.&rft_id=http%3A%2F%2Foeisf.org%2Findex.html%23IPXFER&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://oeis.org">"The On-Line Encyclopedia of Integer Sequences (OEIS)"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+On-Line+Encyclopedia+of+Integer+Sequences+%28OEIS%29&rft_id=https%3A%2F%2Foeis.org&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://oeis.org/FAQ.html#Z27">"FAQ for the On-Line Encyclopedia of Integer Sequences"</a>. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 22 June 2024.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+On-Line+Encyclopedia+of+Integer+Sequences&rft.atitle=FAQ+for+the+On-Line+Encyclopedia+of+Integer+Sequences&rft_id=https%3A%2F%2Foeis.org%2FFAQ.html%23Z27&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"> </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSloane2024" class="citation web cs1">Sloane, Neil (2024). <a rel="nofollow" class="external text" href="https://oeis.org/ol.html">"The Email Servers and Superseeker"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Email+Servers+and+Superseeker&rft.date=2024&rft.aulast=Sloane&rft.aufirst=Neil&rft_id=https%3A%2F%2Foeis.org%2Fol.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"> </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBorwein2017" class="citation book cs1">Borwein, Jonathan M. (2017). "Adventures with the OEIS". In Andrews, George E.; Garvan, Frank (eds.). Analytic Number Theory, Modular Forms and q-Hypergeometric Series. Springer Proceedings in Mathematics & Statistics. Vol. 221. Cham: Springer International Publishing. pp. 123–138. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-68376-8_9">10.1007/978-3-319-68376-8_9</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-68375-1" title="Special:BookSources/978-3-319-68375-1"><bdi>978-3-319-68375-1</bdi></a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2194-1009">2194-1009</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Adventures+with+the+OEIS&rft.btitle=Analytic+Number+Theory%2C+Modular+Forms+and+q-Hypergeometric+Series&rft.place=Cham&rft.series=Springer+Proceedings+in+Mathematics+%26+Statistics&rft.pages=123-138&rft.pub=Springer+International+Publishing&rft.date=2017&rft.issn=2194-1009&rft_id=info%3Adoi%2F10.1007%2F978-3-319-68376-8_9&rft.isbn=978-3-319-68375-1&rft.aulast=Borwein&rft.aufirst=Jonathan+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"> </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGleick1987" class="citation news cs1">Gleick, James (January 27, 1987). <a rel="nofollow" class="external text" href="https://www.nytimes.com/1987/01/27/science/in-a-random-world-he-collects-patterns.html">"In a 'random world,' he collects patterns"</a>. 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C1.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+New+York+Times&rft.atitle=In+a+%27random+world%2C%27+he+collects+patterns&rft.pages=C1&rft.date=1987-01-27&rft.aulast=Gleick&rft.aufirst=James&rft_id=https%3A%2F%2Fwww.nytimes.com%2F1987%2F01%2F27%2Fscience%2Fin-a-random-world-he-collects-patterns.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"> </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> <a rel="nofollow" class="external text" href="http://www.cs.uwaterloo.ca/journals/JIS/">Journal of Integer Sequences</a> (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://www.worldcat.org/search?fq=x0:jrnl&q=n2:1530-7638">1530-7638</a>) </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="http://oeis.org/wiki/Editorial_Board">"Editorial Board"</a>. On-Line Encyclopedia of Integer Sequences.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Editorial+Board&rft.btitle=On-Line+Encyclopedia+of+Integer+Sequences&rft_id=http%3A%2F%2Foeis.org%2Fwiki%2FEditorial_Board&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"> </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeil_Sloane2010" class="citation web cs1">Neil Sloane (2010-11-17). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160207093721/http://oeisf.org/announcementNov2010.txt">"New version of OEIS"</a>. Archived from <a rel="nofollow" class="external text" href="http://oeisf.org/announcementNov2010.txt">the original</a> on 2016-02-07. Retrieved 2011-01-21.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=New+version+of+OEIS&rft.date=2010-11-17&rft.au=Neil+Sloane&rft_id=http%3A%2F%2Foeisf.org%2FannouncementNov2010.txt&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"> </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeil_J._A._Sloane2011" class="citation web cs1">Neil J. A. Sloane (2011-11-14). <a rel="nofollow" class="external text" href="http://list.seqfan.eu/pipermail/seqfan/2011-November/015853.html">"[seqfan] A200000"</a>. SeqFan mailing list. Retrieved 2011-11-22.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=SeqFan+mailing+list&rft.atitle=%5Bseqfan%5D+A200000&rft.date=2011-11-14&rft.au=Neil+J.+A.+Sloane&rft_id=http%3A%2F%2Flist.seqfan.eu%2Fpipermail%2Fseqfan%2F2011-November%2F015853.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"> </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeil_J._A._Sloane2011" class="citation web cs1">Neil J. A. Sloane (2011-11-22). <a rel="nofollow" class="external text" href="http://list.seqfan.eu/pipermail/seqfan/2011-November/015926.html">"[seqfan] A200000 chosen"</a>. SeqFan mailing list. Retrieved 2011-11-22.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=SeqFan+mailing+list&rft.atitle=%5Bseqfan%5D+A200000+chosen&rft.date=2011-11-22&rft.au=Neil+J.+A.+Sloane&rft_id=http%3A%2F%2Flist.seqfan.eu%2Fpipermail%2Fseqfan%2F2011-November%2F015926.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"> </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://oeis.org/wiki/Suggested_Projects">"Suggested Projects"</a>. OEIS wiki. Retrieved 2011-11-22.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=OEIS+wiki&rft.atitle=Suggested+Projects&rft_id=http%3A%2F%2Foeis.org%2Fwiki%2FSuggested_Projects&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"> </li> <li id="cite_note-MATH_VALUES_2023_Sloane-16"><a href="#cite_ref-MATH_VALUES_2023_Sloane_16-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathvalues.org/masterblog/fifty-years-of-integer-sequences">"Fifty Years of Integer Sequences"</a>. MATH VALUES. 2023-12-01. Retrieved 2023-12-04.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MATH+VALUES&rft.atitle=Fifty+Years+of+Integer+Sequences&rft.date=2023-12-01&rft_id=https%3A%2F%2Fwww.mathvalues.org%2Fmasterblog%2Ffifty-years-of-integer-sequences&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"> </li> <li id="cite_note-Sloane_2023_pp._193–205-17"><a href="#cite_ref-Sloane_2023_pp._193%E2%80%93205_17-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSloane2023" class="citation journal cs1">Sloane, N. J. A. (2023). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00283-023-10266-6">""A Handbook of Integer Sequences" Fifty Years Later"</a>. The Mathematical Intelligencer. 45 (3): 193–205. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2301.03149">2301.03149</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00283-023-10266-6">10.1007/s00283-023-10266-6</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0343-6993">0343-6993</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Intelligencer&rft.atitle=%22A+Handbook+of+Integer+Sequences%22+Fifty+Years+Later&rft.volume=45&rft.issue=3&rft.pages=193-205&rft.date=2023&rft_id=info%3Aarxiv%2F2301.03149&rft.issn=0343-6993&rft_id=info%3Adoi%2F10.1007%2Fs00283-023-10266-6&rft.aulast=Sloane&rft.aufirst=N.+J.+A.&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%252Fs00283-023-10266-6&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"> </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://oeis.org/wiki/Welcome#Arrangement_of_the_Sequences_in_Database">"Welcome: Arrangement of the Sequences in Database"</a>. OEIS Wiki. Retrieved 2016-05-05.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=OEIS+Wiki&rft.atitle=Welcome%3A+Arrangement+of+the+Sequences+in+Database&rft_id=https%3A%2F%2Foeis.org%2Fwiki%2FWelcome%23Arrangement_of_the_Sequences_in_Database&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"> </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSloane" class="citation web cs1">Sloane, N. J. A. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180517140606/http://neilsloane.com/doc/sg.pdf">"My favorite integer sequences"</a> (PDF). p. 10. Archived from <a rel="nofollow" class="external text" href="http://neilsloane.com/doc/sg.pdf">the original</a> (PDF) on 2018-05-17.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=My+favorite+integer+sequences&rft.pages=10&rft.aulast=Sloane&rft.aufirst=N.+J.+A.&rft_id=http%3A%2F%2Fneilsloane.com%2Fdoc%2Fsg.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"> </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFN.J.A._Sloane" class="citation web cs1"><a href="/wiki/Neil_Sloane" title="Neil Sloane">N.J.A. Sloane</a>. <a rel="nofollow" class="external text" href="https://oeis.org/eishelp2.html">"Explanation of Terms Used in Reply From"</a>. OEIS.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Explanation+of+Terms+Used+in+Reply+From&rft.pub=OEIS&rft.au=N.J.A.+Sloane&rft_id=https%3A%2F%2Foeis.org%2Feishelp2.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"> </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://oeis.org/wiki/Style_Sheet">"OEIS Style sheet"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=OEIS+Style+sheet&rft_id=http%3A%2F%2Foeis.org%2Fwiki%2FStyle_Sheet&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"> </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://oeis.org/wiki/B-files">"B-Files"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=B-Files&rft_id=http%3A%2F%2Foeis.org%2Fwiki%2FB-files&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"> </li> <li id="cite_note-terms-explanation-23"><a href="#cite_ref-terms-explanation_23-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="http://oeis.org/classic/eishelp2.html">"Explanation of Terms Used in Reply From"</a>. On-Line Encyclopedia of Integer Sequences.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Explanation+of+Terms+Used+in+Reply+From&rft.btitle=On-Line+Encyclopedia+of+Integer+Sequences&rft_id=http%3A%2F%2Foeis.org%2Fclassic%2Feishelp2.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"> </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> The person who submitted A085808 did so as an example of a sequence that should not have been included in the OEIS. Sloane added it anyway, surmising that the sequence "might appear one day on a quiz." </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuglielmetti2008" class="citation web cs1 cs1-prop-foreign-lang-source">Guglielmetti, Philippe (24 August 2008). <a rel="nofollow" class="external text" href="http://www.drgoulu.com/2008/08/24/nombres-acratopeges">"Chasse aux nombres acratopèges"</a>. Pourquoi Comment Combien (in French).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Pourquoi+Comment+Combien&rft.atitle=Chasse+aux+nombres+acratop%C3%A8ges&rft.date=2008-08-24&rft.aulast=Guglielmetti&rft.aufirst=Philippe&rft_id=http%3A%2F%2Fwww.drgoulu.com%2F2008%2F08%2F24%2Fnombres-acratopeges&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"> </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuglielmetti2009" class="citation web cs1 cs1-prop-foreign-lang-source">Guglielmetti, Philippe (18 April 2009). <a rel="nofollow" class="external text" href="http://www.drgoulu.com/2009/04/18/nombres-mineralises">"La minéralisation des nombres"</a>. Pourquoi Comment Combien (in French). Retrieved 25 December 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Pourquoi+Comment+Combien&rft.atitle=La+min%C3%A9ralisation+des+nombres&rft.date=2009-04-18&rft.aulast=Guglielmetti&rft.aufirst=Philippe&rft_id=http%3A%2F%2Fwww.drgoulu.com%2F2009%2F04%2F18%2Fnombres-mineralises&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"> </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGauvritDelahayeZenil2011" class="citation journal cs1">Gauvrit, Nicolas; Delahaye, Jean-Paul; Zenil, Hector (2011). <a rel="nofollow" class="external text" href="https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1048&context=jhm">"Sloane's Gap. Mathematical and Social Factors Explain the Distribution of Numbers in the OEIS"</a>. Journal of Humanistic Mathematics. 3: 3–19. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1101.4470">1101.4470</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2011arXiv1101.4470G">2011arXiv1101.4470G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5642%2Fjhummath.201301.03">10.5642/jhummath.201301.03</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:22115501">22115501</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Humanistic+Mathematics&rft.atitle=Sloane%27s+Gap.+Mathematical+and+Social+Factors+Explain+the+Distribution+of+Numbers+in+the+OEIS&rft.volume=3&rft.pages=3-19&rft.date=2011&rft_id=info%3Aarxiv%2F1101.4470&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A22115501%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.5642%2Fjhummath.201301.03&rft_id=info%3Abibcode%2F2011arXiv1101.4470G&rft.aulast=Gauvrit&rft.aufirst=Nicolas&rft.au=Delahaye%2C+Jean-Paul&rft.au=Zenil%2C+Hector&rft_id=https%3A%2F%2Fscholarship.claremont.edu%2Fcgi%2Fviewcontent.cgi%3Farticle%3D1048%26context%3Djhm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"> </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=_YysNM2JoFo">"Sloane's Gap"</a> (video). <a href="/wiki/Numberphile" title="Numberphile">Numberphile</a>. 2013-10-15. <a rel="nofollow" class="external text" href="https://ghostarchive.org/varchive/youtube/20211117/_YysNM2JoFo">Archived</a> from the original on 2021-11-17. <q>With Dr. James Grime, <a href="/wiki/University_of_Nottingham" title="University of Nottingham">University of Nottingham</a></q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Numberphile&rft.atitle=Sloane%27s+Gap&rft.date=2013-10-15&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3D_YysNM2JoFo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"> </li> </ol></div></div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(9)"><h2 id="References">References</h2> <a role="button" href="/w/index.php?title=On-Line_Encyclopedia_of_Integer_Sequences&action=edit&section=12" title="Edit section: References" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-9 collapsible-block" id="mf-section-9"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBorweinCorless1996" class="citation journal cs1"><a href="/wiki/Jonathan_Borwein" title="Jonathan Borwein">Borwein, J.</a>; Corless, R. 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SIAM Review. 38 (2): 333–337. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F1038058">10.1137/1038058</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SIAM+Review&rft.atitle=The+Encyclopedia+of+Integer+Sequences+%28N.+J.+A.+Sloane+and+Simon+Plouffe%29&rft.volume=38&rft.issue=2&rft.pages=333-337&rft.date=1996&rft_id=info%3Adoi%2F10.1137%2F1038058&rft.aulast=Borwein&rft.aufirst=J.&rft.au=Corless%2C+R.&rft_id=http%3A%2F%2Fwww.cecm.sfu.ca%2F~jborwein%2Fsloane%2Fsloane.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCatchpole2004" class="citation journal cs1">Catchpole, H. (2004). <a rel="nofollow" class="external text" href="http://abc.net.au/science/news/stories/s1209743.htm">"Exploring the number jungle online"</a>. ABC Science. <a href="/wiki/Australian_Broadcasting_Corporation" title="Australian Broadcasting Corporation">Australian Broadcasting Corporation</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=ABC+Science&rft.atitle=Exploring+the+number+jungle+online&rft.date=2004&rft.aulast=Catchpole&rft.aufirst=H.&rft_id=http%3A%2F%2Fabc.net.au%2Fscience%2Fnews%2Fstories%2Fs1209743.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDelarte2004" class="citation journal cs1">Delarte, A. (November 11, 2004). "Mathematician reaches 100k milestone for online integer archive". <a href="/wiki/The_South_End" title="The South End">The South End</a>: 5.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+South+End&rft.atitle=Mathematician+reaches+100k+milestone+for+online+integer+archive&rft.pages=5&rft.date=2004-11-11&rft.aulast=Delarte&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHayes1996" class="citation journal cs1"><a href="/wiki/Brian_Hayes_(scientist)" title="Brian Hayes (scientist)">Hayes, B.</a> (1996). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20151005070524/http://lacim.uqam.ca/~plouffe/articles/A%20Question%20of%20Numbers.pdf">"A Question of Numbers"</a> (PDF). <a href="/wiki/American_Scientist" title="American Scientist">American Scientist</a>. 84 (1): 10–14. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1996AmSci..84...10H">1996AmSci..84...10H</a>. Archived from <a rel="nofollow" class="external text" href="http://lacim.uqam.ca/~plouffe/articles/A%20Question%20of%20Numbers.pdf">the original</a> (PDF) on 2015-10-05. Retrieved 2010-06-01.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Scientist&rft.atitle=A+Question+of+Numbers&rft.volume=84&rft.issue=1&rft.pages=10-14&rft.date=1996&rft_id=info%3Abibcode%2F1996AmSci..84...10H&rft.aulast=Hayes&rft.aufirst=B.&rft_id=http%3A%2F%2Flacim.uqam.ca%2F~plouffe%2Farticles%2FA%2520Question%2520of%2520Numbers.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeterson2003" class="citation journal cs1"><a href="/wiki/Ivars_Peterson" title="Ivars Peterson">Peterson, I.</a> (2003). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170510101432/http://www.plouffe.fr/simon/OEIS/citations/Math%20Trek_%20Sequence%20Puzzles,%20Science%20News%20Online,%20May%2017,%202003.pdf">"Sequence Puzzles"</a> (PDF). <a href="/wiki/Science_News" title="Science News">Science News</a>. 163 (20). Archived from <a rel="nofollow" class="external text" href="http://www.plouffe.fr/simon/OEIS/citations/Math%20Trek_%20Sequence%20Puzzles,%20Science%20News%20Online,%20May%2017,%202003.pdf">the original</a> (PDF) on 2017-05-10. Retrieved 2016-12-24.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science+News&rft.atitle=Sequence+Puzzles&rft.volume=163&rft.issue=20&rft.date=2003&rft.aulast=Peterson&rft.aufirst=I.&rft_id=http%3A%2F%2Fwww.plouffe.fr%2Fsimon%2FOEIS%2Fcitations%2FMath%2520Trek_%2520Sequence%2520Puzzles%2C%2520Science%2520News%2520Online%2C%2520May%252017%2C%25202003.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRehmeyer2010" class="citation journal cs1">Rehmeyer, J. (2010). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131014165834/https://www.sciencenews.org/article/pattern-collector">"The Pattern Collector — Science News"</a>. <a href="/wiki/Science_News" title="Science News">Science News</a>. www.sciencenews.org. Archived from <a rel="nofollow" class="external text" href="https://www.sciencenews.org/article/pattern-collector">the original</a> on 2013-10-14. Retrieved 2010-08-08.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science+News&rft.atitle=The+Pattern+Collector+%E2%80%94+Science+News&rft.date=2010&rft.aulast=Rehmeyer&rft.aufirst=J.&rft_id=https%3A%2F%2Fwww.sciencenews.org%2Farticle%2Fpattern-collector&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"></li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(10)"><h2 id="Further_reading">Further reading</h2> <a role="button" href="/w/index.php?title=On-Line_Encyclopedia_of_Integer_Sequences&action=edit&section=13" title="Edit section: Further reading" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-10 collapsible-block" id="mf-section-10"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRoberts,_S.2023" class="citation cs2">Roberts, S. (May 21, 2023), <a rel="nofollow" class="external text" href="https://www.nytimes.com/2023/05/21/science/math-puzzles-integer-sequences.html">"What Number Comes Next? The Encyclopedia of Integer Sequences Knows."</a>, The New York Times, retrieved 21 May 2023</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+New+York+Times&rft.atitle=What+Number+Comes+Next%3F+The+Encyclopedia+of+Integer+Sequences+Knows.&rft.date=2023-05-21&rft.au=Roberts%2C+S.&rft_id=https%3A%2F%2Fwww.nytimes.com%2F2023%2F05%2F21%2Fscience%2Fmath-puzzles-integer-sequences.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSloane1999" class="citation book cs1"><a href="/wiki/Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (1999). <a rel="nofollow" class="external text" href="http://neilsloane.com/doc/sg.pdf">"My favorite integer sequences"</a> (PDF). In Ding, C.; Helleseth, T.; <a href="/wiki/Harald_Niederreiter" title="Harald Niederreiter">Niederreiter, H.</a> (eds.). Sequences and their Applications (Proceedings of SETA '98). London: Springer-Verlag. pp. 103–130. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0207175">math/0207175</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2002math......7175S">2002math......7175S</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=My+favorite+integer+sequences&rft.btitle=Sequences+and+their+Applications+%28Proceedings+of+SETA+%2798%29&rft.place=London&rft.pages=103-130&rft.pub=Springer-Verlag&rft.date=1999&rft_id=info%3Aarxiv%2Fmath%2F0207175&rft_id=info%3Abibcode%2F2002math......7175S&rft.aulast=Sloane&rft.aufirst=N.+J.+A.&rft_id=http%3A%2F%2Fneilsloane.com%2Fdoc%2Fsg.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSloane2003" class="citation journal cs1"><a href="/wiki/Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (2003). <a rel="nofollow" class="external text" href="https://www.ams.org/notices/200308/comm-sloane.pdf">"The On-Line Encyclopedia of Integer Sequences"</a> (PDF). Notices of the American Mathematical Society. 50 (8): 912–915.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notices+of+the+American+Mathematical+Society&rft.atitle=The+On-Line+Encyclopedia+of+Integer+Sequences&rft.volume=50&rft.issue=8&rft.pages=912-915&rft.date=2003&rft.aulast=Sloane&rft.aufirst=N.+J.+A.&rft_id=https%3A%2F%2Fwww.ams.org%2Fnotices%2F200308%2Fcomm-sloane.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSloanePlouffe1995" class="citation book cs1"><a href="/wiki/Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a>; <a href="/wiki/Simon_Plouffe" title="Simon Plouffe">Plouffe, S.</a> (1995). <a rel="nofollow" class="external text" href="http://oeis.org/book.html">The Encyclopedia of Integer Sequences</a>. San Diego: Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-558630-2" title="Special:BookSources/0-12-558630-2"><bdi>0-12-558630-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Encyclopedia+of+Integer+Sequences&rft.place=San+Diego&rft.pub=Academic+Press&rft.date=1995&rft.isbn=0-12-558630-2&rft.aulast=Sloane&rft.aufirst=N.+J.+A.&rft.au=Plouffe%2C+S.&rft_id=http%3A%2F%2Foeis.org%2Fbook.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZabolotskii2022" class="citation journal cs1">Zabolotskii, A. (2022). <a rel="nofollow" class="external text" href="https://habr.com/en/articles/701208/">"The On-Line Encyclopedia of Integer Sequences in 2021"</a>. Mat. Pros. Series 3. 8: 199–212.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mat.+Pros.&rft.atitle=The+On-Line+Encyclopedia+of+Integer+Sequences+in+2021&rft.volume=8&rft.pages=199-212&rft.date=2022&rft.aulast=Zabolotskii&rft.aufirst=A.&rft_id=https%3A%2F%2Fhabr.com%2Fen%2Farticles%2F701208%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOn-Line+Encyclopedia+of+Integer+Sequences" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBilleyTenner2013" class="citation journal cs1"><a href="/wiki/Sara_Billey" title="Sara Billey">Billey, Sara C.</a>; <a href="/wiki/Bridget_Tenner" title="Bridget Tenner">Tenner, Bridget E.</a> (2013). <a rel="nofollow" class="external text" href="https://www.ams.org/notices/201308/rnoti-p1034.pdf">"Fingerprint databases for theorems"</a> (PDF). Notices of the American Mathematical Society. 60 (8): 1034–1039. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1304.3866">1304.3866</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013arXiv1304.3866B">2013arXiv1304.3866B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fnoti1029">10.1090/noti1029</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14435520">14435520</a>.</cite><span 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id='mw-data-after-content'> <div class="read-more-container"></div> </div> <div id="p-lang"> <h4>Languages</h4> <section> <ul id="p-variants" class="minerva-languages"></ul> <ul class="minerva-languages"><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D9%88%D8%B3%D9%88%D8%B9%D8%A9_%D8%A7%D9%84%D9%85%D8%AA%D8%AA%D8%A7%D9%84%D9%8A%D8%A7%D8%AA_%D8%A7%D9%84%D8%B5%D8%AD%D9%8A%D8%AD%D8%A9_%D8%B9%D9%84%D9%89_%D8%A7%D9%84%D8%A5%D9%86%D8%AA%D8%B1%D9%86%D8%AA" title="موسوعة المتتاليات الصحيحة على الإنترنت – Arabic" lang="ar" hreflang="ar" data-title="موسوعة المتتاليات الصحيحة على الإنترنت" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Tam_%C9%99d%C9%99d_ard%C4%B1c%C4%B1ll%C4%B1qlar%C4%B1n%C4%B1n_ensiklopediyas%C4%B1" title="Tam ədəd ardıcıllıqlarının ensiklopediyası – Azerbaijani" lang="az" hreflang="az" data-title="Tam ədəd ardıcıllıqlarının ensiklopediyası" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences – Catalan" lang="ca" hreflang="ca" data-title="On-Line Encyclopedia of Integer Sequences" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences – Czech" lang="cs" hreflang="cs" data-title="On-Line Encyclopedia of Integer Sequences" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/OEIS" title="OEIS – Danish" lang="da" hreflang="da" data-title="OEIS" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences – German" lang="de" hreflang="de" data-title="On-Line Encyclopedia of Integer Sequences" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences – Greek" lang="el" hreflang="el" data-title="On-Line Encyclopedia of Integer Sequences" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/OEIS" title="OEIS – Spanish" lang="es" hreflang="es" data-title="OEIS" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Retatingebla_Enciklopedio_de_Entjerosinsekvoj" title="Retatingebla Enciklopedio de Entjerosinsekvoj – Esperanto" lang="eo" hreflang="eo" data-title="Retatingebla Enciklopedio de Entjerosinsekvoj" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D8%A7%D9%86%D8%B4%D9%86%D8%A7%D9%85%D9%87_%D8%A8%D8%B1%D8%AE%D8%B7_%D8%AF%D9%86%D8%A8%D8%A7%D9%84%D9%87%E2%80%8C%D9%87%D8%A7%DB%8C_%D8%B5%D8%AD%DB%8C%D8%AD" title="دانشنامه برخط دنباله‌های صحیح – Persian" lang="fa" hreflang="fa" data-title="دانشنامه برخط دنباله‌های صحیح" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Encyclop%C3%A9die_en_ligne_des_suites_de_nombres_entiers" title="Encyclopédie en ligne des suites de nombres entiers – French" lang="fr" hreflang="fr" data-title="Encyclopédie en ligne des suites de nombres entiers" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences – Galician" lang="gl" hreflang="gl" data-title="On-Line Encyclopedia of Integer Sequences" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-inh mw-list-item"><a href="https://inh.wikipedia.org/wiki/%D0%91%D3%80%D0%B0%D1%80%D1%87%D1%87%D0%B0-%D1%82%D0%B0%D1%8C%D1%80%D0%B0%D1%85%D1%8C%D0%B8%D0%B9_%D0%BF%D0%BE%D1%81%D0%BB%D0%B5%D0%B4%D0%BE%D0%B2%D0%B0%D1%82%D0%B5%D0%BB%D1%8C%D0%BD%D0%BE%D1%81%D1%82%D0%B8%D0%B9_%D0%BE%D0%BD%D0%BB%D0%B0%D0%B9%D0%BD-%D1%8D%D0%BD%D1%86%D0%B8%D0%BA%D0%BB%D0%BE%D0%BF%D0%B5%D0%B4%D0%B8" title="БӀарчча-таьрахьий последовательностий онлайн-энциклопеди – Ingush" lang="inh" hreflang="inh" data-title="БӀарчча-таьрахьий последовательностий онлайн-энциклопеди" data-language-autonym="ГӀалгӀай" data-language-local-name="Ingush" class="interlanguage-link-target">ГӀалгӀай</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%98%A8%EB%9D%BC%EC%9D%B8_%EC%A0%95%EC%88%98%EC%97%B4_%EC%82%AC%EC%A0%84" title="온라인 정수열 사전 – Korean" lang="ko" hreflang="ko" data-title="온라인 정수열 사전" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences – Indonesian" lang="id" hreflang="id" data-title="On-Line Encyclopedia of Integer Sequences" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences – Italian" lang="it" hreflang="it" data-title="On-Line Encyclopedia of Integer Sequences" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%90%D7%A0%D7%A6%D7%99%D7%A7%D7%9C%D7%95%D7%A4%D7%93%D7%99%D7%94_%D7%94%D7%9E%D7%A7%D7%95%D7%95%D7%A0%D7%AA_%D7%9C%D7%A1%D7%93%D7%A8%D7%95%D7%AA_%D7%A9%D7%9C_%D7%9E%D7%A1%D7%A4%D7%A8%D7%99%D7%9D_%D7%A9%D7%9C%D7%9E%D7%99%D7%9D" title="האנציקלופדיה המקוונת לסדרות של מספרים שלמים – Hebrew" lang="he" hreflang="he" data-title="האנציקלופדיה המקוונת לסדרות של מספרים שלמים" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences – Hungarian" lang="hu" hreflang="hu" data-title="On-Line Encyclopedia of Integer Sequences" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AA%E0%B5%82%E0%B5%BC%E0%B4%A3%E0%B5%8D%E0%B4%A3%E0%B4%B8%E0%B4%82%E0%B4%96%E0%B5%8D%E0%B4%AF%E0%B4%BE%E0%B4%95%E0%B5%8D%E0%B4%B0%E0%B4%AE%E0%B4%99%E0%B5%8D%E0%B4%99%E0%B4%B3%E0%B5%81%E0%B4%9F%E0%B5%86_%E0%B4%93%E0%B5%BA%E0%B4%B2%E0%B5%88%E0%B5%BB_%E0%B4%B5%E0%B4%BF%E0%B4%9C%E0%B5%8D%E0%B4%9E%E0%B4%BE%E0%B4%A8%E0%B4%95%E0%B5%8B%E0%B4%B6%E0%B4%82" title="പൂർണ്ണസംഖ്യാക്രമങ്ങളുടെ ഓൺലൈൻ വിജ്ഞാനകോശം – Malayalam" lang="ml" hreflang="ml" data-title="പൂർണ്ണസംഖ്യാക്രമങ്ങളുടെ ഓൺലൈൻ വിജ്ഞാനകോശം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences – Dutch" lang="nl" hreflang="nl" data-title="On-Line Encyclopedia of Integer Sequences" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%AA%E3%83%B3%E3%83%A9%E3%82%A4%E3%83%B3%E6%95%B4%E6%95%B0%E5%88%97%E5%A4%A7%E8%BE%9E%E5%85%B8" title="オンライン整数列大辞典 – Japanese" lang="ja" hreflang="ja" data-title="オンライン整数列大辞典" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="On-Line Encyclopedia of Integer Sequences" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences – Polish" lang="pl" hreflang="pl" data-title="On-Line Encyclopedia of Integer Sequences" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/OEIS" title="OEIS – Portuguese" lang="pt" hreflang="pt" data-title="OEIS" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Enciclopedia_electronic%C4%83_a_%C8%99irurilor_de_numere_%C3%AEntregi" title="Enciclopedia electronică a șirurilor de numere întregi – Romanian" lang="ro" hreflang="ro" data-title="Enciclopedia electronică a șirurilor de numere întregi" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9E%D0%BD%D0%BB%D0%B0%D0%B9%D0%BD-%D1%8D%D0%BD%D1%86%D0%B8%D0%BA%D0%BB%D0%BE%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F_%D1%86%D0%B5%D0%BB%D0%BE%D1%87%D0%B8%D1%81%D0%BB%D0%B5%D0%BD%D0%BD%D1%8B%D1%85_%D0%BF%D0%BE%D1%81%D0%BB%D0%B5%D0%B4%D0%BE%D0%B2%D0%B0%D1%82%D0%B5%D0%BB%D1%8C%D0%BD%D0%BE%D1%81%D1%82%D0%B5%D0%B9" title="Онлайн-энциклопедия целочисленных последовательностей – Russian" lang="ru" hreflang="ru" data-title="Онлайн-энциклопедия целочисленных последовательностей" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences – Simple English" lang="en-simple" hreflang="en-simple" data-title="On-Line Encyclopedia of Integer Sequences" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A6%DB%8C%D9%86%D8%B3%D8%A7%DB%8C%DA%A9%DA%B5%DB%86%D9%BE%DB%8C%D8%AF%DB%8C%D8%A7%DB%8C_%D8%B3%DB%95%D8%B1%DA%BE%DB%8E%DA%B5%DB%8C_%D9%BE%D8%A7%D8%B4%DB%8C%DB%95%DA%A9%DB%8C%DB%8C%DB%95_%D8%AA%DB%95%D9%88%D8%A7%D9%88%DB%95%DA%A9%D8%A7%D9%86" title="ئینسایکڵۆپیدیای سەرھێڵی پاشیەکییە تەواوەکان – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ئینسایکڵۆپیدیای سەرھێڵی پاشیەکییە تەواوەکان" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%95%D0%BD%D1%86%D0%B8%D0%BA%D0%BB%D0%BE%D0%BF%D0%B5%D0%B4%D0%B8%D1%98%D0%B0_%D0%BD%D0%B8%D0%B7%D0%BE%D0%B2%D0%B0_%D1%86%D0%B5%D0%BB%D0%B8%D1%85_%D0%B1%D1%80%D0%BE%D1%98%D0%B5%D0%B2%D0%B0_%D0%BD%D0%B0_%D0%BC%D1%80%D0%B5%D0%B6%D0%B8" title="Енциклопедија низова целих бројева на мрежи – Serbian" lang="sr" hreflang="sr" data-title="Енциклопедија низова целих бројева на мрежи" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences – Finnish" lang="fi" hreflang="fi" data-title="On-Line Encyclopedia of Integer Sequences" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/N%C3%A4tuppslagsverket_%C3%B6ver_heltalsf%C3%B6ljder" title="Nätuppslagsverket över heltalsföljder – Swedish" lang="sv" hreflang="sv" data-title="Nätuppslagsverket över heltalsföljder" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A8%E0%AF%87%E0%AE%B0%E0%AE%BF%E0%AE%A3%E0%AF%88%E0%AE%AF_%E0%AE%8E%E0%AE%A3%E0%AF%8D%E0%AE%B5%E0%AE%B0%E0%AE%BF%E0%AE%9A%E0%AF%88_%E0%AE%95%E0%AE%B2%E0%AF%88%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AE%B3%E0%AE%9E%E0%AF%8D%E0%AE%9A%E0%AE%BF%E0%AE%AF%E0%AE%AE%E0%AF%8D" title="நேரிணைய எண்வரிசை கலைக்களஞ்சியம் – Tamil" lang="ta" hreflang="ta" data-title="நேரிணைய எண்வரிசை கலைக்களஞ்சியம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Tamsay%C4%B1_Dizilerinin_%C3%87evrimi%C3%A7i_Ansiklopedisi" title="Tamsayı Dizilerinin Çevrimiçi Ansiklopedisi – Turkish" lang="tr" hreflang="tr" data-title="Tamsayı Dizilerinin Çevrimiçi Ansiklopedisi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%86%D0%BD%D1%82%D0%B5%D1%80%D0%B0%D0%BA%D1%82%D0%B8%D0%B2%D0%BD%D0%B0_%D0%B5%D0%BD%D1%86%D0%B8%D0%BA%D0%BB%D0%BE%D0%BF%D0%B5%D0%B4%D1%96%D1%8F_%D1%86%D1%96%D0%BB%D0%BE%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D0%B8%D1%85_%D0%BF%D0%BE%D1%81%D0%BB%D1%96%D0%B4%D0%BE%D0%B2%D0%BD%D0%BE%D1%81%D1%82%D0%B5%D0%B9" title="Інтерактивна енциклопедія цілочислових послідовностей – Ukrainian" lang="uk" hreflang="uk" data-title="Інтерактивна енциклопедія цілочислових послідовностей" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/B%E1%BA%A3ng_tra_c%E1%BB%A9u_d%C3%A3y_s%E1%BB%91_nguy%C3%AAn_tr%E1%BB%B1c_tuy%E1%BA%BFn" title="Bảng tra cứu dãy số nguyên trực tuyến – Vietnamese" lang="vi" hreflang="vi" data-title="Bảng tra cứu dãy số nguyên trực tuyến" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E6%95%B4%E6%95%B8%E6%95%B8%E5%88%97%E7%B7%9A%E4%B8%8A%E5%A4%A7%E5%85%A8" title="整數數列線上大全 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="整數數列線上大全" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target">文言</a></li><li 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