A363173 Number of triangles inside a regular n-gon formed by intersecting line segments, considering all configurations of 3 line segments from 6 distinct vertices.

0, 0, 0, 0, 7, 16, 84, 180, 462, 796, 1716, 2856, 5005, 7744, 12376, 17508, 27132, 38160, 54264, 73788, 100947, 132216, 177100, 228748, 296010, 374808, 475020, 584140, 736281, 903168, 1107568, 1341232, 1623160, 1939308, 2324784, 2755380, 3262623, 3832080, 4496388

Offset: 3

Author

Paolo Xausa, May 19 2023

Keywords

• nonn

Links

• Paolo Xausa, Table of n, a(n) for n = 3..10000
• Bjorn Poonen and Michael Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG], 1995-2006.
• Steven E. Sommars and Tim Sommars, The Number of Triangles Formed by Intersecting Diagonals of a Regular Polygon, Journal of Integer Sequences, Vol. 1 (1998), Article 98.1.5.
• Paolo Xausa, Illustration of a(9).
• Sequences formed by drawing all diagonals in regular polygon

Crossrefs

Column k = 6 of A363174.

Cf. A000579, A006561, A260417.

Programs

Mathematica

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]

Formula

a(n) = binomial(n,6) = A000579(n) if n is odd, A000579(n) - A260417(n/2) if n is even.