A014626 Number of intersection points of diagonals of an n-gon in general position, plus number of vertices.

0, 1, 2, 3, 5, 10, 21, 42, 78, 135, 220, 341, 507, 728, 1015, 1380, 1836, 2397, 3078, 3895, 4865, 6006, 7337, 8878, 10650, 12675, 14976, 17577, 20503, 23780, 27435, 31496, 35992, 40953, 46410, 52395, 58941, 66082, 73853, 82290, 91430, 101311

Offset: 0

Author

Mohammad K. Azarian

Keywords

• nonn
• easy

Comments

If Y is a 3-subset of an n-set X then, for n >= 4, a(n-3) is the number of 4-subsets of X which have neither one element nor two elements in common with Y; a(n-3) is then also the number of (n-4)-subsets of X which have neither one element nor two elements in common with Y. - Milan Janjic, Dec 28 2007

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Programs

Magma

[(n^4-6*n^3+11*n^2-6*n)/24 +n: n in [0..50]]; // Vincenzo Librandi, Aug 21 2011

Mathematica

Table[(n^4 -6*n^3 +11*n^2 +18*n)/24, {n, 0, 50}] (* G. C. Greubel, Nov 08 2018 *)

PARI

vector(50, n, n--; (n^4 -6*n^3 +11*n^2 +18*n)/24) \\ G. C. Greubel, Nov 08 2018

Formula

a(n) = (n^4 - 6*n^3 + 11*n^2 + 18*n)/24.

From Paul Barry, Sep 23 2004: (Start)

Binomial transform of (0, 1, 0, 0, 1, 0, 0, 0, ...), or g.f. x+x^4.

G.f.: x*(1-3*x+3*x^2)/(1-x)^5;

a(n) = C(n,1) + C(n,4). (End)

E.g.f.: x*(24 + x^3)*exp(x)/24. - G. C. Greubel, Nov 08 2018

Extensions

Corrected and extended by Erich Friedman