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%I #29 Jun 27 2023 10:28:09
%S 0,0,0,0,7,16,84,180,462,796,1716,2856,5005,7744,12376,17508,27132,
%T 38160,54264,73788,100947,132216,177100,228748,296010,374808,475020,
%U 584140,736281,903168,1107568,1341232,1623160,1939308,2324784,2755380,3262623,3832080,4496388
%N Number of triangles inside a regular n-gon formed by intersecting line segments, considering all configurations of 3 line segments from 6 distinct vertices.
%H Paolo Xausa, <a href="/A363173/b363173.txt">Table of n, a(n) for n = 3..10000</a>
%H Bjorn Poonen and Michael Rubinstein, <a href="https://arxiv.org/abs/math/9508209">The number of intersection points made by the diagonals of a regular polygon</a>, arXiv:math/9508209 [math.MG], 1995-2006.
%H Steven E. Sommars and Tim Sommars, <a href="https://cs.uwaterloo.ca/journals/JIS/sommars/newtriangle.html">The Number of Triangles Formed by Intersecting Diagonals of a Regular Polygon</a>, Journal of Integer Sequences, Vol. 1 (1998), Article 98.1.5.
%H Paolo Xausa, <a href="/A363173/a363173.pdf">Illustration of a(9)</a>.
%H <a href="/index/Pol#Poonen">Sequences formed by drawing all diagonals in regular polygon</a>
%F a(n) = binomial(n,6) = A000579(n) if n is odd, A000579(n) - A260417(n/2) if n is even.
%t A363173list[nmax_]:=Module[{d},d[m_,n_]:=Boole[Divisible[n,m]];Table[Binomial[n, 6]-If[EvenQ[n],((1/8n^2-9/8n+7/4)d[2,n]+3/4d[4,n]+(6n-106/3)d[6,n]-33d[12,n]-36d[18,n]-24d[24,n]+96d[30,n]+72d[42,n]+264d[60,n]+96d[84,n]+48d[90,n]+96d[120,n]+48d[210,n])n,0],{n,3,nmax}]];A363173list[50]
%Y Column k = 6 of A363174.
%Y Cf. A000579, A006561, A260417.
%K nonn
%O 3,5
%A _Paolo Xausa_, May 19 2023