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A001106 - 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>A001106 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A001106" 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=%2fA001106">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="A001106 - 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> A001106 </div> <div class=seqname> 9-gonal (or enneagonal or nonagonal) numbers: a(n) = n*(7*n-5)/2. <br><font size=-1>(Formerly M4604)</font> </div> </div> <div class=scorerefs> 87 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>0, 1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 396, 474, 559, 651, 750, 856, 969, 1089, 1216, 1350, 1491, 1639, 1794, 1956, 2125, 2301, 2484, 2674, 2871, 3075, 3286, 3504, 3729, 3961, 4200, 4446, 4699, 4959, 5226, 5500, 5781, 6069, 6364</div> <div class=seqdatalinks> (<a href="/A001106/list">list</a>; <a href="/A001106/graph">graph</a>; <a href="/search?q=A001106+-id:A001106">refs</a>; <a href="/A001106/listen">listen</a>; <a href="/history?seq=A001106">history</a>; <a href="/search?q=id:A001106&fmt=text">text</a>; <a href="/A001106/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,3</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>Sequence found by reading the line from 0, in the direction 0, 9, ... and the parallel line from 1, in the direction 1, 24, ..., in the square spiral whose vertices are the generalized 9-gonal (enneagonal) numbers <a href="/A118277" title="Generalized 9-gonal (or enneagonal) numbers: m*(7*m - 5)/2 with m = 0, 1, -1, 2, -2, 3, -3, ...">A118277</a>. Also sequence found by reading the same lines in the square spiral whose edges have length <a href="/A195019" title="Multiples of 3 and of 4 interleaved: a(2*n-1) = 3*n, a(2*n) = 4*n.">A195019</a> and whose vertices are the numbers <a href="/A195020" title="Vertex number of a square spiral in which the length of the first two edges are the legs of the primitive Pythagorean triple...">A195020</a>. - <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, Sep 10 2011</div> <div class=sectline>Number of ordered pairs of integers (x,y) with abs(x) < n, abs(y) < n and x+y <= n. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Jan 23 2012</div> <div class=sectline>Partial sums give <a href="/A007584" title="9-gonal (or enneagonal) pyramidal numbers: a(n) = n*(n+1)*(7*n-4)/6.">A007584</a>. - <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, Jan 15 2013</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.</div> <div class=sectline>E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.</div> <div class=sectline>N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>T. D. Noe and William A. Tedeschi, <a href="/A001106/b001106.txt">Table of n, a(n) for n = 0..10000</a> (1000 terms were computed by T. D. Noe)</div> <div class=sectline>S. Barbero, U. Cerruti and N. Murru, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barbero/barbero5.html">Transforming Recurrent Sequences by Using the Binomial and Invert Operators</a>, J. Int. Seq. 13 (2010) # 10.7.7, section 4.4.</div> <div class=sectline>C. K. Cook and M. R. Bacon, <a href="https://www.fq.math.ca/Papers1/52-4/CookBacon4292014.pdf">Some polygonal number summation formulas</a>, Fib. Q., 52 (2014), 336-343.</div> <div class=sectline>INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=343">Encyclopedia of Combinatorial Structures 343</a></div> <div class=sectline>Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de s茅ries g茅n茅ratrices et quelques conjectures</a>, Dissertation, Universit茅 du Qu茅bec 脿 Montr茅al, 1992; arXiv:0911.4975 [math.NT], 2009.</div> <div class=sectline>Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992</div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NonagonalNumber.html">Nonagonal Number</a>.</div> <div class=sectline><a href="/index/Pol#polygonal_numbers">Index to sequences related to 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 - 5)*n/2.</div> <div class=sectline>G.f.: x*(1+6*x)/(1-x)^3. - <a href="/wiki/User:Simon_Plouffe">Simon Plouffe</a> in his 1992 dissertation.</div> <div class=sectline>a(n) = n + 7*<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:Floor_van_Lamoen">Floor van Lamoen</a>, Oct 14 2005</div> <div class=sectline>Starting (1, 9, 24, 46, 75, ...) gives the binomial transform of (1, 8, 7, 0, 0, 0, ...). - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Jul 22 2007</div> <div class=sectline>Row sums of triangle <a href="/A131875" title="Triangle, A000012 * A131844 as infinite lower triangular matrices.">A131875</a> starting (1, 9, 24, 46, 75, 111, ...). <a href="/A001106" title="9-gonal (or enneagonal or nonagonal) numbers: a(n) = n*(7*n-5)/2.">A001106</a> = binomial transform of (1, 8, 7, 0, 0, 0, ...). - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Jul 22 2007</div> <div class=sectline>a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(0) = 0, a(1) = 1, a(2) = 9. - <a href="/wiki/User:Jaume_Oliver_Lafont">Jaume Oliver Lafont</a>, Dec 02 2008</div> <div class=sectline>a(n) = 2*a(n-1) - a(n-2) + 7. - <a href="/wiki/User:Mohamed_Bouhamida">Mohamed Bouhamida</a>, May 05 2010</div> <div class=sectline>a(n) = a(n-1) + 7*n - 6 (with a(0) = 0). - <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Nov 12 2010</div> <div class=sectline>a(n) = <a href="/A174738" title="Partial sums of floor(n/7).">A174738</a>(7n). - <a href="/wiki/User:Philippe_Del茅ham">Philippe Del茅ham</a>, Mar 26 2013</div> <div class=sectline>a(7*a(n) + 22*n + 1) = a(7*a(n) + 22*n) + a(7*n+1). - <a href="/wiki/User:Vladimir_Shevelev">Vladimir Shevelev</a>, Jan 24 2014</div> <div class=sectline>E.g.f.: x*(2 + 7*x)*exp(x)/2. - <a href="/wiki/User:Ilya_Gutkovskiy">Ilya Gutkovskiy</a>, Jul 28 2016</div> <div class=sectline>a(n+2) + <a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>(n) = (2*n+3)^2. - <a href="/wiki/User:Ezhilarasu_Velayutham">Ezhilarasu Velayutham</a>, Mar 18 2020</div> <div class=sectline>Product_{n>=2} (1 - 1/a(n)) = 7/9. - <a href="/wiki/User:Amiram_Eldar">Amiram Eldar</a>, Jan 21 2021</div> <div class=sectline>Sum_{n>=1} 1/a(n) = <a href="/A244646" title="Decimal expansion of the sum of the reciprocals of the 9-gonal (or enneagonal or nonagonal) numbers (A001106).">A244646</a>. - <a href="/wiki/User:Amiram_Eldar">Amiram Eldar</a>, Nov 12 2021</div> <div class=sectline>a(n) = <a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>(3*n-2) - (n-1)^2. - <a href="/wiki/User:Charlie_Marion">Charlie Marion</a>, Feb 27 2022</div> <div class=sectline>a(n) = 3*<a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>(n) + 2*<a href="/A005563" title="a(n) = n*(n+2) = (n+1)^2 - 1.">A005563</a>(n-2). In general, if P(k,n) = the n-th k-gonal number, then P(m*k,n) = m*P(k,n) + (m-1)*<a href="/A005563" title="a(n) = n*(n+2) = (n+1)^2 - 1.">A005563</a>(n-2). - <a href="/wiki/User:Charlie_Marion">Charlie Marion</a>, Feb 21 2023</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Table[n(7n - 5)/2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 1, 9}, 50] (* <a href="/wiki/User:Harvey_P._Dale">Harvey P. Dale</a>, Nov 06 2011 *)</div> <div class=sectline>(* For Mathematica 10.4+ *) Table[PolygonalNumber[RegularPolygon[9], n], {n, 0, 43}] (* <a href="/wiki/User:Arkadiusz_Wesolowski">Arkadiusz Wesolowski</a>, Aug 27 2016 *)</div> <div class=sectline>PolygonalNumber[9, Range[0, 50]] (* Requires Mathematica version 10 or later *) (* <a href="/wiki/User:Harvey_P._Dale">Harvey P. Dale</a>, Nov 19 2019 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) a(n)=n*(7*n-5)/2 \\ <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Jun 10 2011</div> <div class=sectline>(Haskell)</div> <div class=sectline>a001106 n = length [(x, y) | x <- [-n+1..n-1], y <- [-n+1..n-1], x + y <= n]</div> <div class=sectline>-- <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Jan 23 2012</div> <div class=sectline>(Haskell) a001106 n = n*(7*n-5) `div` 2 -- <a href="/wiki/User:James_Spahlinger">James Spahlinger</a>, Oct 18 2012</div> <div class=sectline>(Python 3)</div> <div class=sectline>def aList(): # Intended to compute the initial segment of the sequence, not isolated terms.</div> <div class=sectline> x, y = 1, 1</div> <div class=sectline> yield 0</div> <div class=sectline> while True:</div> <div class=sectline> yield x</div> <div class=sectline> x, y = x + y + 7, y + 7</div> <div class=sectline><a href="/A001106" title="9-gonal (or enneagonal or nonagonal) numbers: a(n) = n*(7*n-5)/2.">A001106</a> = aList()</div> <div class=sectline>print([next(<a href="/A001106" title="9-gonal (or enneagonal or nonagonal) numbers: a(n) = n*(7*n-5)/2.">A001106</a>) for i in range(49)]) # <a href="/wiki/User:Peter_Luschny">Peter Luschny</a>, Aug 04 2019</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A093564" title="(7,1) Pascal triangle.">A093564</a> ((7, 1) Pascal, column m=2). Partial sums of <a href="/A016993" title="a(n) = 7*n + 1.">A016993</a>.</div> <div class=sectline>Cf. <a href="/A131875" title="Triangle, A000012 * A131844 as infinite lower triangular matrices.">A131875</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="/A069099" title="Centered heptagonal numbers.">A069099</a>, <a href="/A244646" title="Decimal expansion of the sum of the reciprocals of the 9-gonal (or enneagonal or nonagonal) numbers (A001106).">A244646</a>.</div> <div class=sectline>Sequence in context: <a href="/A067725" title="a(n) = 3*n^2 + 6*n.">A067725</a> <a href="/A213903" title="Least number x such that there are n numbers of the form 6k-1 or 6k+1 between prime(x) and prime(x+1).">A213903</a> <a href="/A351043" title="Lexicographically earliest Racetrack trajectory (using von Neumann neighborhood) on spiral on infinite square grid.">A351043</a> * <a href="/A023551" title="Self-convolution of natural numbers >= 3.">A023551</a> <a href="/A022787" title="Place where n-th 1 occurs in A023125.">A022787</a> <a href="/A365190" title="The weak Schur numbers for 2-coloring.">A365190</a></div> <div class=sectline>Adjacent sequences: <a href="/A001103" title="Numbers k such that (k / product of digits of k) is 1 or a prime.">A001103</a> <a href="/A001104" title="Numbers n such that n / product of digits of n is a square.">A001104</a> <a href="/A001105" title="a(n) = 2*n^2.">A001105</a> * <a href="/A001107" title="10-gonal (or decagonal) numbers: a(n) = n*(4*n-3).">A001107</a> <a href="/A001108" title="a(n)-th triangular number is a square: a(n+1) = 6*a(n) - a(n-1) + 2, with a(0) = 0, a(1) = 1.">A001108</a> <a href="/A001109" title="a(n)^2 is a triangular number: a(n) = 6*a(n-1) - a(n-2) with a(0)=0, a(1)=1.">A001109</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></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>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 November 27 18:22 EST 2024. Contains 378168 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>