A190400 Number of ways to place 5 nonattacking grasshoppers on a toroidal chessboard of size n x n.

0, 0, 0, 976, 18510, 201528, 1232448, 5637824, 20502396, 63720920, 174647286, 434439792, 997037470, 2141831160, 4348204020, 8412482304, 15605151496, 27903377016, 48291880442, 81188237680, 132977239290, 212739639640

Offset

1,4

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)

Formula

Explicit formula: a(n) = 1/120*n^2*(n^8 -10*n^6 -240*n^5 +995*n^4 +640*n^3 +1870*n^2 -41680*n +69624) + 2*n^2*(n-3)*(2n+1)*(-1)^n, n>6.

G.f.: 2*x^4*(144*x^18 -874*x^17 +1356*x^16 +2195*x^15 -8778*x^14 +4282*x^13 +16170*x^12 -23696*x^11 -5686*x^10 +36079*x^9 -33008*x^8 -33909*x^7 -13310*x^6 -61448*x^5 -197358*x^4 -109070*x^3 -50114*x^2 -6327*x -488)/((x-1)^11*(x+1)^5).

Mathematica

CoefficientList[Series[2 x^3 (144 x^18 - 874 x^17 + 1356 x^16 + 2195 x^15 - 8778 x^14 + 4282 x^13 + 16170 x^12 - 23696 x^11 - 5686 x^10 + 36079 x^9 - 33008 x^8 - 33909 x^7 - 13310 x^6 - 61448 x^5 - 197358 x^4 - 109070 x^3 - 50114 x^2 - 6327 x- 488) / ((x - 1)^11 (x + 1)^5), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 03 2013 *)

Crossrefs

Cf. A190397, A190398, A190399, A173775.

Sequence in context: A253907 A233984 A205252 * A288917 A264130 A282405

Adjacent sequences: A190397 A190398 A190399 * A190401 A190402 A190403

Keyword

nonn,easy

Author

Vaclav Kotesovec, May 10 2011

Status

approved