A190398 Number of ways to place 3 nonattacking grasshoppers on a toroidal chessboard of size n x n.

0, 4, 72, 496, 2100, 6708, 17640, 40384, 83376, 158900, 284108, 482160, 783484, 1227156, 1862400, 2750208, 3965080, 5596884, 7752836, 10559600, 14165508, 18742900, 24490584, 31636416, 40440000, 51195508, 64234620, 79929584

Offset: 1

Author

Vaclav Kotesovec, May 10 2011

Keywords

• nonn
• easy

Comments

The Grasshopper moves on the same lines as a queen, but must jump over a hurdle to land on the square immediately beyond.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights (in English and Czech)

Crossrefs

Cf.: A190395, A172518.

Programs

Mathematica

CoefficientList[Series[- 4 x (3 x^8 - 17 x^7 + 37 x^6 - 35 x^5 + 11 x^4 + 19 x^2 + 11 x + 1) / (x - 1)^7, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 02 2013 *)

Formula

a(n) = 1/6*n^2*(n^4 -3*n^2 -24*n +74), n>3.

G.f.: -4*x^2*(3*x^8 -17*x^7 +37*x^6 -35*x^5 +11*x^4 +19*x^2 +11*x +1)/(x-1)^7.