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 A014628 Number of segments (and sides) created by diagonals of an n-gon in general position. 1
 3, 8, 20, 45, 91, 168, 288, 465, 715, 1056, 1508, 2093, 2835, 3760, 4896, 6273, 7923, 9880, 12180, 14861, 17963, 21528, 25600, 30225, 35451, 41328, 47908, 55245, 63395, 72416, 82368, 93313, 105315, 118440, 132756, 148333, 165243, 183560 (list; graph; refs; listen; history; text; internal format)
 OFFSET 3,1 COMMENTS There is a connection to A014626: number of intersection points of diagonals of n-gon, plus number of vertices, b(n) = n*(n+1)*(n^2-7*n+18)/24 and A006522: number of regions created by sides and diagonals of n-gon, c(n) = (n-1)*(n-2)*(n^2-3*n+12)/24. These are related by the Euler-formula: b(n) + c(n) - a(n) = 1. - Georg Wengler, Mar 31 2005 LINKS Table of n, a(n) for n=3..40. Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1). FORMULA a(n) = (n^4-6*n^3+17*n^2-24*n)/12 + n; or equally n*(n-1)*(n^2-5*n+12)/12. G.f.: x^3*(3-7*x+10*x^2-5*x^3+x^4)/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009 a(n) = C(n,2) + 2*C(n,4). - Gary Detlefs, Jun 06 2010 MATHEMATICA Table[Binomial[n, 2]+2Binomial[n, 4], {n, 3, 50}] (* Harvey P. Dale, Oct 03 2020 *) CROSSREFS Cf. A014626, A006522, A134393. Sequence in context: A096585 A057765 A290866 * A134393 A034504 A191522 Adjacent sequences: A014625 A014626 A014627 * A014629 A014630 A014631 KEYWORD nonn,easy AUTHOR Mohammad K. Azarian EXTENSIONS G.f. proposed by Maksym Voznyy, checked and corrected by R. J. Mathar, Sep 16 2009 More terms from Erich Friedman Offset corrected by Mohammad K. Azarian, Nov 19 2008 Offset corrected by Eric Rowland, Aug 15 2017 STATUS approved

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Last modified September 17 23:36 EDT 2024. Contains 375991 sequences. (Running on oeis4.)