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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
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
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
KEYWORD
nonn,easy
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