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%I M4513 #46 Jun 18 2021 11:37:08
%S 1,8,35,110,287,632,1302,2400,4257,6956,11297,17234,25935,37424,53516,
%T 73404,101745,136200,181279,236258,306383,389264,495650,620048,772785,
%U 951384,1167453,1410350,1716191,2058848,2463384,2924000,3462305,4067028,4776219,5568786,6479551
%N Total number of triangles visible in regular n-gon with all diagonals drawn.
%C Place n equally-spaced points on a circle, join them in all possible ways; how many triangles can be seen?
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A006600/b006600.txt">Table of n, a(n) for n=3..1000</a>
%H Sascha Kurz, <a href="http://www.mathe2.uni-bayreuth.de/sascha/oeis/drawing/drawing.html">m-gons in regular n-gons</a>
%H Victor Meally, <a href="/A006556/a006556.pdf">Letter to N. J. A. Sloane</a>, no date.
%H B. Poonen and M. Rubinstein, <a href="http://epubs.siam.org:80/sam-bin/dbq/article/28124">Number of Intersection Points Made by the Diagonals of a Regular Polygon</a>, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.
%H B. Poonen and M. Rubinstein, <a href="http://math.mit.edu/~poonen/papers/ngon.pdf">The number of intersection points made by the diagonals of a regular polygon</a>, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
%H B. Poonen and M. Rubinstein, <a href="http://arXiv.org/abs/math.MG/9508209">The number of intersection points made by the diagonals of a regular polygon</a>, arXiv version, which has fewer typos than the SIAM version.
%H B. Poonen and M. Rubinstein, <a href="http://math.mit.edu/~poonen/papers/ngon.m">Mathematica programs for these sequences</a>
%H M. Rubinstein, <a href="/A006561/a006561_3.pdf">Drawings for n=4,5,6,...</a>
%H T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/triangle_counting">Number of triangles for convex n-gon</a>
%H S. E. Sommars and T. Sommars, <a href="http://www.cs.uwaterloo.ca/journals/JIS/sommars/newtriangle.html">Number of Triangles Formed by Intersecting Diagonals of a Regular Polygon</a>, J. Integer Sequences, 1 (1998), #98.1.5.
%H <a href="/index/Pol#Poonen">Sequences formed by drawing all diagonals in regular polygon</a>
%F a(2n-1) = A005732(2n-1) for n > 1; a(2n) = A005732(2n) - A260417(n) for n > 1. - _Jonathan Sondow_, Jul 25 2015
%e a(4) = 8 because in a quadrilateral the diagonals cross to make four triangles, which pair up to make four more.
%t del[m_,n_]:=If[Mod[n,m]==0,1,0]; Tri[n_]:=n(n-1)(n-2)(n^3+18n^2-43n+60)/720 - del[2,n](n-2)(n-7)n/8 - del[4,n](3n/4) - del[6,n](18n-106)n/3 + del[12,n]*33n + del[18,n]*36n + del[24,n]*24n - del[30,n]*96n - del[42,n]*72n - del[60,n]*264n - del[84,n]*96n - del[90,n]*48n - del[120,n]*96n - del[210,n]*48n; Table[Tri[n], {n,3,1000}] (* _T. D. Noe_, Dec 21 2006 *)
%Y Often confused with A005732.
%Y Cf. A203016, A260417.
%Y Sequences related to chords in a circle: A001006, A054726, A006533, A006561, A006600, A007569, A007678. See also entries for chord diagrams in Index file.
%K nonn,easy,nice
%O 3,2
%A _N. J. A. Sloane_
%E a(3)-a(8) computed by Victor Meally (personal communication to _N. J. A. Sloane_, circa 1975); later terms and recurrence from S. Sommars and T. Sommars.
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