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 A006600 Total number of triangles visible in regular n-gon with all diagonals drawn. (Formerly M4513) 16
 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 (list; graph; refs; listen; history; text; internal format)
 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 D. Radcliffe, Counting triangles in a regular polygon 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. 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 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

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Last modified November 14 19:12 EST 2018. Contains 317214 sequences. (Running on oeis4.)