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A005891 - 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>A005891 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A005891" 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=%2fA005891">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="A005891 - 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> A005891 </div> <div class=seqname> Centered pentagonal numbers: (5n^2+5n+2)/2; crystal ball sequence for 3.3.3.4.4. planar net. <br><font size=-1>(Formerly M4112)</font> </div> </div> <div class=scorerefs> 85 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 6, 16, 31, 51, 76, 106, 141, 181, 226, 276, 331, 391, 456, 526, 601, 681, 766, 856, 951, 1051, 1156, 1266, 1381, 1501, 1626, 1756, 1891, 2031, 2176, 2326, 2481, 2641, 2806, 2976, 3151, 3331, 3516, 3706, 3901, 4101, 4306, 4516, 4731, 4951, 5176, 5406</div> <div class=seqdatalinks> (<a href="/A005891/list">list</a>; <a href="/A005891/graph">graph</a>; <a href="/search?q=A005891+-id:A005891">refs</a>; <a href="/A005891/listen">listen</a>; <a href="/history?seq=A005891">history</a>; <a href="/search?q=id:A005891&fmt=text">text</a>; <a href="/A005891/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,2</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>Equals the triangular numbers convolved with [1, 3, 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>From <a href="/wiki/User:Ant_King">Ant King</a>, Jun 15 2012: (Start)</div> <div class=sectline>a(n) == 1 (mod 5) for all n.</div> <div class=sectline>The digital roots of the a(n) form a purely periodic palindromic 9-cycle 1, 6, 7, 4, 6, 4, 7, 6, 1.</div> <div class=sectline>The units' digits of the a(n) form a purely periodic palindromic 4-cycle 1, 6, 6, 1.</div> <div class=sectline>(End)</div> <div class=sectline>Binomial transform of (1, 5, 5, 0, 0, 0, ...) and second partial sum of (1, 4, 5, 5, 5, ...). - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Sep 09 2015</div> <div class=sectline>a(n) = a(-1-n) for all n in Z. - <a href="/wiki/User:Michael_Somos">Michael Somos</a>, Jan 25 2019</div> <div class=sectline>On the plane start with a single regular pentagon, and repeat the following procedure, "For each edge of any pentagon not already connected to an existing pentagon create a mirror image such that the mirror image does not overlap with an existing pentagon." a(n) is the number of pentagons occupying the plane after n repetitions. - <a href="/wiki/User:Torlach_Rush">Torlach Rush</a>, Sep 14 2022</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <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>B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>T. D. Noe, <a href="/A005891/b005891.txt">Table of n, a(n) for n = 0..1000</a></div> <div class=sectline>Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.</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>Cliff Reiter, <a href="http://hdl.handle.net/10385/2864">Polygonal Numbers and Fifty One Stars</a>, Lafayette College, Easton, PA (2019).</div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CenteredPentagonalNumber.html">Centered Pentagonal Number.</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 class=sectline><a href="/index/Cor#crystal_ball">Index entries for crystal ball sequences</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>G.f.: (1 + 3*x + x^2)/(1 - x)^3. <a href="/wiki/User:Simon_Plouffe">Simon Plouffe</a> in his 1992 dissertation</div> <div class=sectline>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, 5, 0, 0, 0, ...]. - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Dec 29 2007</div> <div class=sectline>a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(0)=1, a(1)=6, a(2)=16. - <a href="/wiki/User:Jaume_Oliver_Lafont">Jaume Oliver Lafont</a>, Dec 02 2008</div> <div class=sectline>a(n) = 5*<a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>(n) + 1 = 5*T(n) + 1, for n = 0, 1, 2, 3, ... and where T(n) = n*(n+1)/2 = n-th triangular number. - <a href="/wiki/User:Thomas_M._Green">Thomas M. Green</a>, Nov 25 2009</div> <div class=sectline>a(n) = a(n-1) + 5*n, with a(0)=1. - <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Nov 18 2010</div> <div class=sectline>a(n) = <a href="/A028895" title="5 times triangular numbers: a(n) = 5*n*(n+1)/2.">A028895</a>(n) + 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) + 5. - <a href="/wiki/User:Ant_King">Ant King</a>, Jun 12 2012</div> <div class=sectline>Sum_{n>=0} 1/a(n) = 2*Pi /sqrt(15) *tanh(Pi/2*sqrt(3/5)) = 1.360613169863... - <a href="/wiki/User:Ant_King">Ant King</a>, Jun 15 2012</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>(5,n). - <a href="/wiki/User:R._J._Mathar">R. J. Mathar</a>, Jul 28 2016</div> <div class=sectline>E.g.f.: (2 + 10*x + 5*x^2)*exp(x)/2. - <a href="/wiki/User:Ilya_Gutkovskiy">Ilya Gutkovskiy</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>=0} a(n)/n! = 17*e/2.</div> <div class=sectline>Sum_{n>=0} (-1)^(n+1)*a(n)/n! = 3/(2*e). (End)</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>a(2)= 5*T(2) + 1 = 5*3 + 1 = 16, a(4) = 5*T(4) + 1 = 5*10 + 1 = 51. - <a href="/wiki/User:Thomas_M._Green">Thomas M. Green</a>, Nov 16 2009</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline><a href="/A005891" title="Centered pentagonal numbers: (5n^2+5n+2)/2; crystal ball sequence for 3.3.3.4.4. planar net.">A005891</a> := proc(n)</div> <div class=sectline> 1+5*n*(1+n)/2 ;</div> <div class=sectline>end proc:</div> <div class=sectline>seq(<a href="/A005891" title="Centered pentagonal numbers: (5n^2+5n+2)/2; crystal ball sequence for 3.3.3.4.4. planar net.">A005891</a>(n), n=0..40) ; # <a href="/wiki/User:R._J._Mathar">R. J. Mathar</a>, Oct 07 2021</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>FoldList[#1 + #2 &, 1, 5 Range@ 40] (* <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, 6, 16}, 50] (* <a href="/wiki/User:Harvey_P._Dale">Harvey P. Dale</a>, Sep 08 2018 *)</div> <div class=sectline>Table[ j! Coefficient[Series[Exp[x]*(1 + 5 x^2/2)-1, {x, 0, 20}], x, j], {j, 0, 20}] (* <a href="/wiki/User:Nikolaos_Pantelidis">Nikolaos Pantelidis</a>, Feb 07 2023 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) a(n)=5*n*(n+1)/2+1 \\ <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Mar 22 2016</div> <div class=sectline>(Magma) [5*n*(n+1)/2 + 1: n in [0..50]]; // <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Nov 04 2017</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A028895" title="5 times triangular numbers: a(n) = 5*n*(n+1)/2.">A028895</a>, <a href="/A001844" title="Centered square numbers: a(n) = 2*n*(n+1)+1. Sums of two consecutive squares. Also, consider all Pythagorean triples (X, Y, ...">A001844</a>, <a href="/A003215" title="Hex (or centered hexagonal) numbers: 3*n*(n+1)+1 (crystal ball sequence for hexagonal lattice).">A003215</a>, <a href="/A004068" title="Number of atoms in a decahedron with n shells.">A004068</a> (partial sums), <a href="/A006322" title="4-dimensional analog of centered polygonal numbers.">A006322</a>, <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>.</div> <div class=sectline>Partial sums of <a href="/A008706" title="Coordination sequence for 3.3.3.4.4 planar net.">A008706</a>.</div> <div class=sectline>Equals second row of <a href="/A167546" title="The ED1 array read by antidiagonals">A167546</a> divided by 2.</div> <div class=sectline>Sequence in context: <a href="/A102214" title="Expansion of (1 + 4*x + 4*x^2)/((1+x)*(1-x)^3).">A102214</a> <a href="/A301679" title="Partial sums of A301678.">A301679</a> <a href="/A115007" title="Row 3 of array in A114999.">A115007</a> * <a href="/A108182" title="Cumulative sum of antisquares (A080255).">A108182</a> <a href="/A244242" title="Number of partitions of n into 6 parts such that every i-th smallest part (counted with multiplicity) is different from i.">A244242</a> <a href="/A092286" title="Fourth diagonal (m=3) of triangle A084938; a(n) = A084938(n+3,n) = (n^3 + 9*n^2 + 26*n)/6.">A092286</a></div> <div class=sectline>Adjacent sequences: <a href="/A005888" title="Theta series of hexagonal close-packing with respect to edge between layers.">A005888</a> <a href="/A005889" title="Theta series of hexagonal close-packing with respect to triangle between octahedra.">A005889</a> <a href="/A005890" title="Theta series of hexagonal close-packing with respect to center of triangle between two layers.">A005890</a> * <a href="/A005892" title="Truncated square numbers: 7*n^2 + 4*n + 1.">A005892</a> <a href="/A005893" title="Number of points on surface of tetrahedron; coordination sequence for sodalite net (equals 2*n^2+2 for n > 0).">A005893</a> <a href="/A005894" title="Centered tetrahedral numbers.">A005894</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></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>Formula corrected and relocated by <a href="/wiki/User:Johannes_W._Meijer">Johannes W. Meijer</a>, Nov 07 2009</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 25 07:13 EST 2024. 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