A006600 Total number of triangles visible in regular n-gon with all diagonals drawn.

1, 8, 35, 110, 287, 632, 1302, 2400, 4257, 6956, 11297, 17234, 25935, 37424, 53516, 73404, 101745, 136200, 181279, 236258, 306383, 389264, 495650, 620048, 772785, 951384, 1167453, 1410350, 1716191, 2058848, 2463384, 2924000, 3462305, 4067028, 4776219, 5568786, 6479551

Offset: 3

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Place n equally-spaced points on a circle, join them in all possible ways; how many triangles can be seen?

Examples

			a(4) = 8 because in a quadrilateral the diagonals cross to make four triangles, which pair up to make four more.
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n=3..1000
• Sascha Kurz, m-gons in regular n-gons
• Victor Meally, Letter to N. J. A. Sloane, no date.
• B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.
• B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
• B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv version, which has fewer typos than the SIAM version.
• B. Poonen and M. Rubinstein, Mathematica programs for these sequences
• M. Rubinstein, Drawings for n=4,5,6,...
• T. Sillke, Number of triangles for convex n-gon
• S. E. Sommars and T. Sommars, Number of Triangles Formed by Intersecting Diagonals of a Regular Polygon, J. Integer Sequences, 1 (1998), #98.1.5.
• Sequences formed by drawing all diagonals in regular polygon

Crossrefs

Often confused with A005732.

Cf. A203016, A260417.

Row sums of A363174.

Sequences related to chords in a circle: A001006, A054726, A006533, A006561, A006600, A007569, A007678. See also entries for chord diagrams in Index file.

Programs

Mathematica

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, Dec 21 2006 *)

Formula

a(2n-1) = A005732(2n-1) for n > 1; a(2n) = A005732(2n) - A260417(n) for n > 1. - Jonathan Sondow, Jul 25 2015

Extensions

a(3)-a(8) computed by Victor Meally (personal communication to N. J. A. Sloane, circa 1975); later terms and recurrence from S. Sommars and T. Sommars.