A190395 -id:A190395 - cp's OEIS frontend

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

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.: A190395, A061994.

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 *)

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.

A190398 Number of ways to place 3 nonattacking grasshoppers on a toroidal chessboard of size n x n.

Original entry on oeis.org

0, 4, 72, 496, 2100, 6708, 17640, 40384, 83376, 158900, 284108, 482160, 783484, 1227156, 1862400, 2750208, 3965080, 5596884, 7752836, 10559600, 14165508, 18742900, 24490584, 31636416, 40440000, 51195508, 64234620, 79929584

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.: A190395, A172518.

CoefficientList[Series[- 4 x (3 x^8 - 17 x^7 + 37 x^6 - 35 x^5 + 11 x^4 + 19 x^2 + 11 x + 1) / (x - 1)^7, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 02 2013 *)

a(n) = 1/6*n^2*(n^4 -3*n^2 -24*n +74), n>3.

G.f.: -4*x^2*(3*x^8 -17*x^7 +37*x^6 -35*x^5 +11*x^4 +19*x^2 +11*x +1)/(x-1)^7.

A190397 Number of ways to place 5 nonattacking grasshoppers on a chessboard of size n x n.

Original entry on oeis.org

0, 0, 28, 1668, 29092, 252584, 1441634, 6222996, 22004086, 66972760, 181332416, 446905476, 1019470032, 2179712872, 4410518630, 8510498516, 15756224370, 28128603736, 48622240660, 81660504068, 133643402268, 213660267432

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. A190395, A190396, A108792.

CoefficientList[Series[2 x^2 (8 x^12 - 60 x^11 + 75 x^10 + 24 x^9 + 441 x^8 - 1948 x^7 - 893 x^6 + 4122 x^5 - 8491 x^4 - 15988 x^3 - 6822 x^2 - 694 x - 14) / ((x - 1)^11 (x+1)), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 02 2013 *)

a(n) = 1/120*(n^10 -10*n^8 -200*n^7 +1175*n^6 -1136*n^5 -740*n^4 -30520*n^3 +159624*n^2 -289024*n +179175 -135*(-1)^n), n>3.

G.f.: 2x^3*(8*x^12 -60*x^11 +75*x^10 +24*x^9 +441*x^8 -1948*x^7 -893*x^6 +4122*x^5 -8491*x^4 -15988*x^3 -6822*x^2 -694*x -14)/((x-1)^11*(x+1)).

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).

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

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A190398 Number of ways to place 3 nonattacking grasshoppers on a toroidal chessboard of size n x n.

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A190397 Number of ways to place 5 nonattacking grasshoppers on a chessboard of size n x n.

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

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