A190396 Number of ways to place 4 nonattacking grasshoppers on a chessboard of size n x n.

0, 1, 78, 1278, 10002, 50191, 189208, 584958, 1563488, 3737987, 8181786, 16669638, 32003238, 58438623, 102234772, 172344406, 281269668, 446107043, 689807558, 1042679982, 1544166426, 2244921423, 3209227248, 4517779918

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, A061994.

Programs

Mathematica

CoefficientList[Series[- x (2 x^9 - 22 x^8 + 50 x^7 + 78 x^6 - 89 x^5 - 245 x^4 + 1224 x^3 + 612 x^2 + 69 x + 1) / (x - 1)^9, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 02 2013 *)

Formula

a(n) = 1/24*(n^8 -6*n^6 -80*n^5 +431*n^4 -552*n^3 -666*n^2 +2168*n -1392), n>2.

G.f.: -x^2*(2*x^9 -22*x^8 +50*x^7 +78*x^6 -89*x^5 -245*x^4 +1224*x^3 +612*x^2 +69*x +1)/(x-1)^9.