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A006534 - OEIS
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <html> <head> <link rel="stylesheet" href="/styles.css"> <meta name="format-detection" content="telephone=no"> <meta http-equiv="content-type" content="text/html; charset=utf-8"> <meta name=viewport content="width=device-width, initial-scale=1"> <meta name="keywords" content="OEIS,integer sequences,Sloane" /> <title>A006534 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A006534" function redir() { var host = document.location.hostname; if(host != "oeis.org" && host != "127.0.0.1" && !/^([0-9.]+)$/.test(host) && host != "localhost" && host != "localhost.localdomain") { document.location = "https"+":"+"//"+"oeis"+".org/" + myURL; } } function sf() { if(document.location.pathname == "/" && document.f) document.f.q.focus(); } </script> </head> <body bgcolor=#ffffff onload="redir();sf()"> <div class=loginbar> <div class=login> <a href="/login?redirect=%2fA006534">login</a> </div> </div> <div class=center><div class=top> <center> <div class=donors> The OEIS is supported by <a href="http://oeisf.org/#DONATE">the many generous donors to the OEIS Foundation</a>. </div> <div class=banner> <a href="/"><img class=banner border="0" width="600" src="/banner2021.jpg" alt="A006534 - OEIS"></a> </div> </center> </div></div> <div class=center><div class=pagebody> <div class=searchbarcenter> <form name=f action="/search" method="GET"> <div class=searchbargreet> <div class=searchbar> <div class=searchq> <input class=searchbox maxLength=1024 name=q value="" title="Search Query"> </div> <div class=searchsubmit> <input type=submit value="Search" name=go> </div> <div class=hints> <span class=hints><a href="/hints.html">Hints</a></span> </div> </div> <div class=searchgreet> (Greetings from <a href="/welcome">The On-Line Encyclopedia of Integer Sequences</a>!) </div> </div> </form> </div> <div class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> A006534 </div> <div class=seqname> Number of one-sided triangular polyominoes (n-iamonds) with n cells; turning over not allowed, holes are allowed. <br><font size=-1>(Formerly M3287)</font> </div> </div> <div class=scorerefs> 7 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 1, 1, 4, 6, 19, 43, 120, 307, 866, 2336, 6588, 18373, 52119, 147700, 422016, 1207477, 3471067, 9999135, 28893560, 83665729, 242826187, 706074369, 2056870697, 6001555275, 17538335077, 51323792789, 150390053432, 441210664337, 1295886453860, 3810208448847, 11214076720061, 33035788241735</div> <div class=seqdatalinks> (<a href="/A006534/list">list</a>; <a href="/A006534/graph">graph</a>; <a href="/search?q=A006534+-id:A006534">refs</a>; <a href="/A006534/listen">listen</a>; <a href="/history?seq=A006534">history</a>; <a href="/search?q=id:A006534&fmt=text">text</a>; <a href="/A006534/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>1,4</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>The figures are formed by connecting n regular triangles by edges.</div> <div class=sectline>"Turning over not allowed" means that axial symmetric polyiamonds are counted separately, thus a(4) = 4 and a(5) = 6 while <a href="/A000577" title="Number of triangular polyominoes (or triangular polyforms, or polyiamonds) with n cells (turning over is allowed, holes are ...">A000577</a>(4) = 3 and <a href="/A000577" title="Number of triangular polyominoes (or triangular polyforms, or polyiamonds) with n cells (turning over is allowed, holes are ...">A000577</a>(5) = 4, cf. examples. - <a href="/wiki/User:M._F._Hasler">M. F. Hasler</a>, Nov 12 2017</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>F. Harary, Graphical enumeration problems; in Graph Theory and Theoretical Physics, ed. F. Harary, Academic Press, London, 1967, pp. 1-41.</div> <div class=sectline>W. F. Lunnon, Counting hexagonal and triangular polyominoes, pp. 87-100 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.</div> <div class=sectline>N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).</div> <div class=sectline>P. J. Torbijn, Polyiamonds, J. Rec. Math. 2 (1969), 216-227.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>John Mason, <a href="/A006534/b006534.txt">Table of n, a(n) for n = 1..52</a></div> <div class=sectline>R. K. Guy, <a href="http://www.g4g4.com/contentsmmpp.html">O'Beirne's Hexiamond</a>, in The Mathemagician and the Pied Puzzler - A Collection in Tribute to Martin Gardner, Ed. E. R. Berlekamp and T. Rogers, A. K. Peters, 1999, 85-96.</div> <div class=sectline>J. Meeus & N. J. A. Sloane, <a href="/A006532/a006532_1.pdf">Correspondence, 1974-1975</a></div> <div class=sectline>Ed Pegg, Jr., <a href="http://demonstrations.wolfram.com/PolyformExplorer/">Illustrations of polyforms</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Polyiamond.html">Polyiamond.</a></div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>From <a href="/wiki/User:M._F._Hasler">M. F. Hasler</a>, Nov 12 2017: (Start)</div> <div class=sectline>Putting dots for the approximate center of the regular triangles (alternatively flipped up and down for neighboring dots), we have:</div> <div class=sectline>a(4) = #{ .... , .:. , ..: , :.. } = 4, while ..: and :.. are considered equivalent and not counted twice in <a href="/A000577" title="Number of triangular polyominoes (or triangular polyforms, or polyiamonds) with n cells (turning over is allowed, holes are ...">A000577</a>(4) = 3.</div> <div class=sectline>a(5) = #{ ..... , ...: , :... , ..:. , .:.. , :.: } = 6, and again the 2nd & 3rd and 4th & 5th are considered equivalent and not counted twice in <a href="/A000577" title="Number of triangular polyominoes (or triangular polyforms, or polyiamonds) with n cells (turning over is allowed, holes are ...">A000577</a>(5) = 4. (End)</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A000577" title="Number of triangular polyominoes (or triangular polyforms, or polyiamonds) with n cells (turning over is allowed, holes are ...">A000577</a> (same with "turning over allowed"), <a href="/A030223" title="Number of achiral triangular n-ominoes (n-iamonds) (holes are allowed).">A030223</a>, <a href="/A030224" title="Triangular polyominoes (n-iamonds) without bilateral symmetry (holes are allowed).">A030224</a>, <a href="/A001420" title="Number of fixed 2-dimensional triangular-celled animals with n cells (n-iamonds, polyiamonds) in the 2-dimensional hexagonal...">A001420</a>.</div> <div class=sectline>Sequence in context: <a href="/A343677" title="Number of (2n+1)-digit undulating alternating palindromic primes.">A343677</a> <a href="/A222379" title="Number of distinct functions f representable as x -> x^x^...^x with n x's and parentheses inserted in all possible ways givi...">A222379</a> <a href="/A202618" title="a(n) is the smallest integer that is the sum of n distinct terms of A075058.">A202618</a> * <a href="/A064035" title="Erroneous version of A006534.">A064035</a> <a href="/A010364" title="Class A multi-edge stars with n edges and 2 odd labeled roots.">A010364</a> <a href="/A110391" title="a(n) = L(3*n)/L(n), where L(n) = Lucas number.">A110391</a></div> <div class=sectline>Adjacent sequences: <a href="/A006531" title="Semiorders on n elements.">A006531</a> <a href="/A006532" title="Numbers whose sum of divisors is a square.">A006532</a> <a href="/A006533" title="Place n equally-spaced points around a circle and join every pair of points by a chord; this divides the circle into a(n) re...">A006533</a> * <a href="/A006535" title="Number of one-sided hexagonal polyominoes with n cells.">A006535</a> <a href="/A006536" title="Switching classes of digraphs.">A006536</a> <a href="/A006537" title="Worst cases for Pierce expansions (numerators).">A006537</a></div> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <div class=sectline><span title="a sequence of nonnegative numbers">nonn</span>,<span title="next term not known, may be hard to find. please extend this sequence">hard</span>,<span title="an exceptionally nice sequence">nice</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a></div> </div> </div> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <div class=sectline>Corrected and extended by <a href="/wiki/User:David_W._Wilson">David W. Wilson</a></div> <div class=sectline>a(19) from <a href="/wiki/User:Achim_Flammenkamp">Achim Flammenkamp</a>, Feb 15 1999</div> <div class=sectline>a(20) to a(28) from <a href="/wiki/User:Joseph_Myers">Joseph Myers</a>, Sep 24 2002</div> <div class=sectline>Edited by <a href="/wiki/User:M._F._Hasler">M. F. Hasler</a>, Nov 12 2017</div> <div class=sectline>More terms from <a href="/wiki/User:John_Mason">John Mason</a>, Oct 28 2023</div> </div> </div> <div class=section> <div class=sectname>STATUS</div> <div class=sectbody> <div class=sectline>approved</div> </div> </div> </div> <div class=space10></div> </div> </div></div> <p> <div class=footerpad></div> <div class=footer> <center> <div class=bottom> <div class=linksbar> <a href="/">Lookup</a> <a href="/wiki/Welcome"><font color="red">Welcome</font></a> <a href="/wiki/Main_Page"><font color="red">Wiki</font></a> <a href="/wiki/Special:RequestAccount">Register</a> <a href="/play.html">Music</a> <a href="/plot2.html">Plot 2</a> <a href="/demo1.html">Demos</a> <a href="/wiki/Index_to_OEIS">Index</a> <a href="/webcam">WebCam</a> <a href="/Submit.html">Contribute</a> <a href="/eishelp2.html">Format</a> <a href="/wiki/Style_Sheet">Style Sheet</a> <a href="/transforms.html">Transforms</a> <a href="/ol.html">Superseeker</a> <a href="/recent">Recents</a> </div> <div class=linksbar> <a href="/community.html">The OEIS Community</a> </div> <div class=linksbar> Maintained by <a href="http://oeisf.org">The OEIS Foundation Inc.</a> </div> <div class=dbinfo>Last modified March 13 22:06 EDT 2025. Contains 381739 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>