The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A006600 Total number of triangles visible in regular n-gon with all diagonals drawn. (Formerly M4513) 20
 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 10 04:30 EDT 2024. Contains 375773 sequences. (Running on oeis4.)