A190397 -id:A190397 - cp's OEIS frontend

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

Vaclav Kotesovec, May 10 2011

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

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)

Cf. A190397, A190398, A190399, A173775.

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 *)

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).

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

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, May 13 2011

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

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)

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

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 *)

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).

cp's OEIS Frontend

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

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