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A069099 - 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>A069099 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A069099" 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=%2fA069099">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="A069099 - 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> A069099 </div> <div class=seqname> Centered heptagonal numbers. </div> </div> <div class=scorerefs> 63 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 8, 22, 43, 71, 106, 148, 197, 253, 316, 386, 463, 547, 638, 736, 841, 953, 1072, 1198, 1331, 1471, 1618, 1772, 1933, 2101, 2276, 2458, 2647, 2843, 3046, 3256, 3473, 3697, 3928, 4166, 4411, 4663, 4922, 5188, 5461, 5741, 6028, 6322, 6623, 6931, 7246</div> <div class=seqdatalinks> (<a href="/A069099/list">list</a>; <a href="/A069099/graph">graph</a>; <a href="/search?q=A069099+-id:A069099">refs</a>; <a href="/A069099/listen">listen</a>; <a href="/history?seq=A069099">history</a>; <a href="/search?q=id:A069099&fmt=text">text</a>; <a href="/A069099/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>1,2</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>Equals the triangular numbers convolved with [ 1, 5, 1, 0, 0, 0, ...]. - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a> and <a href="/wiki/User:Alexander_R._Povolotsky">Alexander R. Povolotsky</a>, May 29 2009</div> <div class=sectline>Number of ordered pairs of integers (x,y) with abs(x) < n, abs(y) < n and abs(x + y) < n, counting twice pairs of equal numbers. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Jan 23 2012; corrected and extended by <a href="/wiki/User:Mauro_Fiorentini">Mauro Fiorentini</a>, Jan 01 2018</div> <div class=sectline>The number of pairs without repetitions is a(n) - 2n + 3 for n > 1. For example, there are 19 such pairs for n = 3: (-2, 0), (-2, 1), (-2, 2), (-1, -1), (-1, 0), (-1, 1), (-1, 2), (0, -2), (0, -1), (0, 0), (0, 1), (0, 2), (1, -2), (1, -1), (1, 0), (1, 1), (2, -2), (2, -1), (2, 0). - <a href="/wiki/User:Mauro_Fiorentini">Mauro Fiorentini</a>, Jan 01 2018</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>T. D. Noe, <a href="/A069099/b069099.txt">Table of n, a(n) for n = 1..1000</a></div> <div class=sectline>Nicolas B臎lohoubek and Viktor Javor, <a href="/A069099/a069099.pdf">Centered heptagonal numbers appearing in aperiodic heptagonal tiling</a></div> <div class=sectline>Leo Tavares, <a href="/A069099/a069099.jpg">Illustration: Crystal Numbers</a></div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CenteredPolygonalNumber.html">Centered Polygonal Numbers</a>.</div> <div class=sectline><a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a></div> <div class=sectline><a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).</div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = (7*n^2 - 7*n + 2)/2.</div> <div class=sectline>a(n) = 1 + Sum_{k=1..n} 7*k. - Xavier Acloque, Oct 26 2003</div> <div class=sectline>Binomial transform of [1, 7, 7, 0, 0, 0, ...]; Narayana transform (<a href="/A001263" title="Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.">A001263</a>) of [1, 7, 0, 0, 0, ...]. - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Dec 29 2007</div> <div class=sectline>a(n) = 7*n + a(n-1) - 7 (with a(1)=1). - <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Aug 08 2010</div> <div class=sectline>G.f.: x*(1+5*x+x^2) / (1-x)^3. - <a href="/wiki/User:R._J._Mathar">R. J. Mathar</a>, Feb 04 2011</div> <div class=sectline>a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=8, a(2)=22. - <a href="/wiki/User:Harvey_P._Dale">Harvey P. Dale</a>, Jun 04 2011</div> <div class=sectline>a(n) = <a href="/A024966" title="7 times triangular numbers: 7*n*(n+1)/2.">A024966</a>(n-1) + 1. - <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, Oct 03 2011</div> <div class=sectline>a(n) = 2*a(n-1) - a(n-2) + 7. - <a href="/wiki/User:Ant_King">Ant King</a>, Jun 17 2012</div> <div class=sectline>From <a href="/wiki/User:Ant_King">Ant King</a>, Jun 17 2012: (Start)</div> <div class=sectline>Sum_{n>=1} 1/a(n) = 2*Pi/sqrt(7)*tanh(Pi/(2*sqrt(7))) = 1.264723171685652...</div> <div class=sectline>a(n) == 1 (mod 7) for all n.</div> <div class=sectline>The sequence of digital roots of the a(n) is period 9: repeat [1, 8, 4, 7, 8, 7, 4, 8, 1] (the period is a palindrome).</div> <div class=sectline>The sequence of a(n) mod 10 is period 20: repeat [1, 8, 2, 3, 1, 6, 8, 7, 3, 6, 6, 3, 7, 8, 6, 1, 3, 2, 8, 1] (the period is a palindrome).</div> <div class=sectline>(End)</div> <div class=sectline>E.g.f.: -1 + (2 + 7*x^2)*exp(x)/2. - <a href="/wiki/User:Ilya_Gutkovskiy">Ilya Gutkovskiy</a>, Jun 30 2016</div> <div class=sectline>a(n) = <a href="/A101321" title="Table T(n,m) = 1 + n*m*(m+1)/2 read by antidiagonals: centered polygonal numbers.">A101321</a>(7,n-1). - <a href="/wiki/User:R._J._Mathar">R. J. Mathar</a>, Jul 28 2016</div> <div class=sectline>From <a href="/wiki/User:Amiram_Eldar">Amiram Eldar</a>, Jun 20 2020: (Start)</div> <div class=sectline>Sum_{n>=1} a(n)/n! = 9*e/2 - 1.</div> <div class=sectline>Sum_{n>=1} (-1)^n * a(n)/n! = 9/(2*e) - 1. (End)</div> <div class=sectline>a(n) = <a href="/A003215" title="Hex (or centered hexagonal) numbers: 3*n*(n+1)+1 (crystal ball sequence for hexagonal lattice).">A003215</a>(n-1) + <a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>(n-1). - <a href="/wiki/User:Leo_Tavares">Leo Tavares</a>, Jul 19 2022</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>a(5) = 71 because 71 = (7*5^2 - 7*5 + 2)/2 = (175 - 35 + 2)/2 = 142/2.</div> <div class=sectline>From <a href="/wiki/User:Bruno_Berselli">Bruno Berselli</a>, Oct 27 2017: (Start)</div> <div class=sectline>1 = -(0) + (1).</div> <div class=sectline>8 = -(0+1) + (2+3+4).</div> <div class=sectline>22 = -(0+1+2) + (3+4+5+6+7).</div> <div class=sectline>43 = -(0+1+2+3) + (4+5+6+7+8+9+10).</div> <div class=sectline>71 = -(0+1+2+3+4) + (5+6+7+8+9+10+11+12+13). (End)</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>FoldList[#1 + #2 &, 1, 7 Range@ 50] (* <a href="/wiki/User:Robert_G._Wilson_v">Robert G. Wilson v</a>, Feb 02 2011 *)</div> <div class=sectline>LinearRecurrence[{3, -3, 1}, {1, 8, 22}, 50] (* <a href="/wiki/User:Harvey_P._Dale">Harvey P. Dale</a>, Jun 04 2011 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(Haskell)</div> <div class=sectline>a069099 n = length</div> <div class=sectline> [(x, y) | x <- [-n+1..n-1], y <- [-n+1..n-1], x + y <= n - 1]</div> <div class=sectline>-- <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Jan 23 2012</div> <div class=sectline>(PARI) a(n)=(7*n^2-7*n+2)/2 \\ <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Sep 24 2015</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A000566" title="Heptagonal numbers (or 7-gonal numbers): n*(5*n-3)/2.">A000566</a> (heptagonal numbers).</div> <div class=sectline>Cf. <a href="/A001263" title="Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.">A001263</a>, <a href="/A057655" title="The circle problem: number of points (x,y) in square lattice with x^2 + y^2 <= n.">A057655</a>, <a href="/A001106" title="9-gonal (or enneagonal or nonagonal) numbers: a(n) = n*(7*n-5)/2.">A001106</a>.</div> <div class=sectline>Cf. <a href="/A003215" title="Hex (or centered hexagonal) numbers: 3*n*(n+1)+1 (crystal ball sequence for hexagonal lattice).">A003215</a>, <a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>.</div> <div class=sectline>Sequence in context: <a href="/A058508" title="McKay-Thompson series of class 15A for Monster.">A058508</a> <a href="/A134783" title="McKay-Thompson series of class 15A for the Monster group with a(0) = 1.">A134783</a> <a href="/A211529" title="Number of -1..1 arrays x(i) of n+1 elements i=1..n+1 with set{t,u,v in 0,1}((x[i+t]+x[j+u]+x[k+v])*(-1)^(t+u+v)) having two,...">A211529</a> * <a href="/A172473" title="a(n) = floor(sqrt(2*n^5)).">A172473</a> <a href="/A145067" title="Zero followed by partial sums of A008865.">A145067</a> <a href="/A112684" title="Similar to A112683, but using the recurrence x(i) = x(i-1)^2 + x(i-n) mod 3.">A112684</a></div> <div class=sectline>Adjacent sequences: <a href="/A069096" title="Numbers k such that the numerator of Sum_{d|k} 1/d > 3*k.">A069096</a> <a href="/A069097" title="Moebius transform of A064987, n*sigma(n).">A069097</a> <a href="/A069098" title="Number of minimal monic annihilator polynomials over the ring of integers modulo n.">A069098</a> * <a href="/A069100" title="n-th n-digit prime number.">A069100</a> <a href="/A069101" title="a(1) = 2; a(2) = 2; a(n) = Prime[a(n-1) + a(n-2)] if n > 2.">A069101</a> <a href="/A069102" title="a(1) = 1; a(2) = 1; a(n) = Prime[n-1] + Prime[n-2] if n > 2.">A069102</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="it is very easy to produce terms of sequence">easy</span>,<span title="an exceptionally nice sequence">nice</span>,<span title="edited within the last two weeks">changed</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Terrel_Trotter,_Jr.">Terrel Trotter, Jr.</a>, Apr 05 2002</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 February 17 19:03 EST 2025. Contains 380975 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>