A190399 Number of ways to place 4 nonattacking grasshoppers on a toroidal chessboard of size n x n.

0, 1, 54, 1068, 8550, 45873, 177968, 562032, 1519560, 3662625, 8057390, 16477020, 31712850, 58018793, 101639700, 171525568, 280160068, 444636297, 687881890, 1040201500, 1541008350, 2240952065, 3204279960, 4511682288

Offset

1,3

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

a(n) = 1/24*n^2*(n^6 -6*n^4 -96*n^3 +347*n^2 +96*n -726 +96*(-1)^n), n>4.

G.f.: -x^2*(80*x^14 -444*x^13 +768*x^12 +108*x^11 -1824*x^10 +1600*x^9 +1025*x^8 -1200*x^7 +708*x^6 +1772*x^5 +7254*x^4 +2788*x^3 +756*x^2 +48*x +1)/((x-1)^9*(x+1)^3).

Mathematica

CoefficientList[Series[- x (80 x^14 - 444 x^13 + 768 x^12 + 108 x^11 - 1824 x^10 + 1600 x^9 + 1025 x^8 - 1200 x^7 + 708 x^6 + 1772 x^5 + 7254 x^4 + 2788 x^3 + 756 x^2 + 48 x + 1) / ((x - 1)^9 (x + 1)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 03 2013 *)

Crossrefs

Cf. A190396, A190398, A172519.

Sequence in context: A245832 A121625 A341940 * A071803 A215836 A160289

Adjacent sequences: A190396 A190397 A190398 * A190400 A190401 A190402

Keyword

nonn,easy

Author

Vaclav Kotesovec, May 10 2011

Status

approved