A014628 - OEIS

%I #31 Oct 09 2020 17:25:39

%S 3,8,20,45,91,168,288,465,715,1056,1508,2093,2835,3760,4896,6273,7923,

%T 9880,12180,14861,17963,21528,25600,30225,35451,41328,47908,55245,

%U 63395,72416,82368,93313,105315,118440,132756,148333,165243,183560

%N Number of segments (and sides) created by diagonals of an n-gon in general position.

%C 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

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F 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.

%F 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

%F a(n) = C(n,2) + 2*C(n,4). - _Gary Detlefs_, Jun 06 2010

%t Table[Binomial[n,2]+2Binomial[n,4],{n,3,50}] (* _Harvey P. Dale_, Oct 03 2020 *)

%Y Cf. A014626, A006522, A134393.

%K nonn,easy

%O 3,1

%A _Mohammad K. Azarian_

%E G.f. proposed by Maksym Voznyy, checked and corrected by _R. J. Mathar_, Sep 16 2009

%E More terms from _Erich Friedman_

%E Offset corrected by _Mohammad K. Azarian_, Nov 19 2008

%E Offset corrected by _Eric Rowland_, Aug 15 2017