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

0,3

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).

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

Mathematica

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

Prog

(Magma) [(n^4-6*n^3+11*n^2-6*n)/24 +n: n in [0..50]]; // Vincenzo Librandi, Aug 21 2011
(PARI) vector(50, n, n--; (n^4 -6*n^3 +11*n^2 +18*n)/24) \\ G. C. Greubel, Nov 08 2018

Crossrefs

Sequence in context: A023170 A125312 A300550 * A132418 A024494 A131708

Adjacent sequences: A014623 A014624 A014625 * A014627 A014628 A014629

Keyword

nonn,easy

Author

Mohammad K. Azarian

Extensions

Corrected and extended by Erich Friedman

Status

approved