%I A006600 M4513
%S A006600 1,8,35,110,287,632,1302,2400,4257,6956,11297,17234,25935,37424,53516,
               73404,
%T A006600 101745,136200,181279,236258,306383,389264,495650,620048,772785,951384,
               1167453,
%U A006600 1410350,1716191,2058848,2463384,2924000,3462305,4067028,4776219,5568786,
               6479551
%N A006600 Triangles in regular n-gon.
%C A006600 Place n equally-spaced points on a circle, join them in all possible 
               ways; how many triangles can be seen?
%D A006600 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A006600 T. D. Noe, <a href="b006600.txt">Table of n, a(n) for n=3..1000</a>
%H A006600 Sascha Kurz, <a href="http://www.mathe2.uni-bayreuth.de/sascha/oeis/drawing/
               drawing.html">m-gons in regular n-gons</a>
%H A006600 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 A006600 B. Poonen and M. Rubinstein, <a href="http://math.mit.edu/~poonen/">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 A006600 B. Poonen and M. Rubinstein, <a href="http://arXiv.org/pdf/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 A006600 B. Poonen and M. Rubinstein, <a href="http://math.mit.edu/~poonen/papers/
               ngon.m">Mathematica programs for these sequences</a>
%H A006600 D. Radcliffe, <a href="http://gotmath.com/ngon.html">Counting triangles 
               in a regular polygon</a>
%H A006600 T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/
               triangle_counting">Number of triangles for convex n-gon</a>
%H A006600 S. E. Sommars and T. Sommars, <a href="http://www.cs.uwaterloo.ca/journals/
               JIS/index.html">Number of Triangles Formed by Intersecting Diagonals 
               of a Regular Polygon</a>, J. Integer Sequences, 1 (1998), #98.1.5.
%H A006600 <a href="Sindx_Pol.html#Poonen">Sequences formed by drawing all diagonals 
               in regular polygon</a>
%e A006600 a(4) = 8 because in a quadrilateral the diagonals cross to make four 
               triangles, which pair up to make four more.
%t A006600 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 (noe(AT)sspectra.com), Dec 21 2006
%Y A006600 Often confused with A005732.
%Y A006600 Sequences related to chords in a circle: A001006, A054726, A006533, A006561, 
               A006600, A007569, A007678. See also entries for chord diagrams in 
               Index file.
%Y A006600 Sequence in context: A136016 A100907 A058102 this_sequence A005732 A040977 
               A036598
%Y A006600 Adjacent sequences: A006597 A006598 A006599 this_sequence A006601 A006602 
               A006603
%K A006600 nonn,easy,nice
%O A006600 3,2
%A A006600 N. J. A. Sloane (njas(AT)research.att.com).
%E A006600 a(3)...a(8) computed by Victor Meally (personal communication); later 
               terms and recurrence from S. Sommars and T. Sommars.