A014628 Number of segments (and sides) created by diagonals of an n-gon in general position.

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

Author

Mohammad K. Azarian

Keywords

• nonn
• easy

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

• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A014626, A006522, A134393.

Programs

Mathematica

Table[Binomial[n,2]+2Binomial[n,4],{n,3,50}] (* Harvey P. Dale, Oct 03 2020 *)

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

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