A363174 Array read by rows: T(n,k) is the number of triangles inside a regular n-gon formed by intersecting line segments, considering all configurations of 3 line segments from k distinct vertices, with n >= 3, 3 <= k <= 6.

1, 0, 0, 0, 4, 4, 0, 0, 10, 20, 5, 0, 20, 60, 30, 0, 35, 140, 105, 7, 56, 280, 280, 16, 84, 504, 630, 84, 120, 840, 1260, 180, 165, 1320, 2310, 462, 220, 1980, 3960, 796, 286, 2860, 6435, 1716, 364, 4004, 10010, 2856, 455, 5460, 15015, 5005, 560, 7280, 21840, 7744

Offset: 3

Author

Paolo Xausa, May 19 2023

Keywords

• nonn
• tabf

Comments

See Sommars and Sommars (1998) for a complete analysis of the problem.

Examples

			Array begins:
  n\k|     3     4     5     6
  ---+---------------------------
   3 |     1,    0,    0,    0;
   4 |     4,    4,    0,    0;
   5 |    10,   20,    5,    0;
   6 |    20,   60,   30,    0;
   7 |    35,  140,  105,    7;
   8 |    56,  280,  280,   16;
   9 |    84,  504,  630,   84;
  10 |   120,  840, 1260,  180;
  ...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 3..10002 (rows 3..2502 of array, flattened).
• 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 T(9,6).
• Sequences formed by drawing all diagonals in regular polygon

Crossrefs

Cf. A000579, A006561, A006600 (row sums), A260417.

Cf. A000292 (column k = 3), A033488 (column k = 4), A174002 (column k = 5), A363173 (column k = 6).

Programs

Mathematica

A363174list[rowmax_]:=Module[{d},d[m_,n_]:=Boole[Divisible[n,m]];Table[Binomial[n,k]If[4<=k<=5,k,1]-If[k==6&&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,rowmax},{k,3,6}]];A363174list[20]

Formula

T(n,3) = binomial(n,3) = A000292(n-2).

T(n,4) = 4*binomial(n,4) = A033488(n-3).

T(n,5) = 5*binomial(n,5) = A174002(n-4), for n >= 4.

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

Sum_{k=3..6} T(n,k) = A006600(n).