A190579 Number of ways to place 6 nonattacking grasshoppers on an n x n chessboard.

0, 0, 2, 998, 51618, 873852, 8039322, 50272978, 240764814, 947860554, 3210392210, 9649651136, 26316155354, 66191981440, 155482089002, 344411086374, 725043524246, 1459722296638, 2825136685698, 5278863810724, 9557560367842

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) = n^12/720 -n^10/48 -5n^9/9 +509n^8/144 -187n^7/90 +701n^6/48 -14467n^5/36 +666917n^4/360 -471121n^3/180 -59875n^2/24 +57101n/6 -11339/2 -(9n^2/8-n-7/2)*(-1)^n, n>5.

G.f.: 2x^3*(8x^18 -59x^17 +110x^16 +71x^15 +473x^14 -3017x^13 -5401x^12 +23838x^11 -2727x^10 -119474x^9 -45545x^8 -20157x^7 -571677x^6 -1006961x^5 -689547x^4 -199704x^3 -20861x^2 -489x -1)/((x-1)^13*(x+1)^3).

Mathematica

CoefficientList[Series[2 x^2 (8 x^18 - 59 x^17 + 110 x^16 + 71 x^15 + 473 x^14 - 3017 x^13 - 5401 x^12 + 23838 x^11 - 2727 x^10 - 119474 x^9 - 45545 x^8 - 20157 x^7 - 571677 x^6 - 1006961 x^5 - 689547 x^4 - 199704 x^3 - 20861 x^2 - 489 x - 1) / ((x - 1)^13 (x + 1)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 03 2013 *)

Crossrefs

Cf. A190395, A190396, A190397, A176186, A190401.

Sequence in context: A172386 A079233 A194191 * A271527 A214543 A258661

Adjacent sequences: A190576 A190577 A190578 * A190580 A190581 A190582

Keyword

nonn,easy

Author

Vaclav Kotesovec, May 13 2011

Status

approved