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

3,5

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

Formula

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

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]

Crossrefs

Column k = 6 of A363174.

Cf. A000579, A006561, A260417.

Sequence in context: A215180 A309476 A192376 * A214904 A029498 A351531

Adjacent sequences: A363170 A363171 A363172 * A363174 A363175 A363176

Keyword

nonn

Author

Paolo Xausa, May 19 2023

Status

approved