A006600 Total number of triangles visible in regular n-gon with all diagonals drawn. (Formerly M4513)

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

3,2

Comments

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

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

Formula

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

Example

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

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 *)

Crossrefs

Often confused with A005732.

Cf. A203016, A260417.

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

Sequence in context: A189592 A285737 A244882 * A161456 A005732 A162211

Adjacent sequences: A006597 A006598 A006599 * A006601 A006602 A006603

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

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.

Status

approved