A194651 -id:A194651 - cp's OEIS frontend

A194652 Number of ways to place 4 nonattacking kings on an n X n cylindrical chessboard.

0, 0, 0, 32, 1205, 13260, 74494, 291708, 908973, 2416410, 5711530, 12327414, 24743693, 46797968, 84216990, 145288600, 241697109, 389546478, 610597338, 933745570, 1396771845, 2048393204, 2950649438, 4181658708, 5838778525, 8042209890, 10939084074, 14708073198

Offset: 1

Vaclav Kotesovec, Aug 31 2011

nonn

V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights
Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Cf. A061997, A179424, A194650, A194651.

CoefficientList[Series[x^3*(54*x^9 - 384*x^8 + 1052*x^7 - 1263*x^6 + 657*x^5 - 1434*x^4 + 4154*x^3 - 3567*x^2 - 917*x - 32)/(x - 1)^9, {x, 0, 30}], x] (* Wesley Ivan Hurt, Dec 27 2023 *)

a(n) = 1/24*n*(n^7 - 54*n^5 + 36*n^4 + 1019*n^3 - 1236*n^2 - 6690*n + 10884), n>=5.

G.f.: x^4*(54*x^9 - 384*x^8 + 1052*x^7 - 1263*x^6 + 657*x^5 - 1434*x^4 + 4154*x^3 - 3567*x^2 - 917*x - 32)/(x-1)^9.

A194653 Number of ways to place 5 nonattacking kings on an n X n cylindrical chessboard.

Original entry on oeis.org

0, 0, 0, 0, 655, 26952, 309869, 1998752, 9124848, 33065040, 101473009, 274593648, 673080928, 1522931256, 3224953725, 6458355776, 12330557912, 22588294464, 39908439249, 68290845520, 113579839128, 184145882536, 291764365485, 452734505952, 689287992800

Offset: 1

Vaclav Kotesovec, Aug 31 2011

nonn

V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights
Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Cf. A061998, A179425, A194650, A194651, A194652.

a(n) = 1/120*n*(n^9 - 90*n^7 + 60*n^6 + 3155*n^5 - 3900*n^4 - 50910*n^3 + 86580*n^2 + 318864*n - 656160), n>=6.

G.f.: -x^5*(185*x^11 - 1635*x^10 + 6336*x^9 - 15496*x^8 + 32185*x^7 - 62315*x^6 + 86237*x^5 - 49559*x^4 - 35522*x^3 + 49422*x^2 + 19747*x + 655)/(x-1)^11.

A194654 Number of ways to place 6 nonattacking kings on an n X n cylindrical chessboard.

Original entry on oeis.org

0, 0, 0, 0, 125, 28930, 809368, 9414152, 66305781, 338374560, 1378426060, 4751038284, 14388638901, 39296604844, 98605016040, 230507248912, 507379370525, 1060395103800, 2118303772332, 4066797540820, 7537144196589, 13535598398916, 23628635128024, 40203393674520

Offset: 1

Vaclav Kotesovec, Aug 31 2011

nonn

V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights
Index entries for linear recurrences with constant coefficients, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).

Cf. A172158, A179426, A194650, A194651, A194652, A194653.

a(n) = 1/720*n*(n^11 - 135*n^9 + 90*n^8 + 7525*n^7 - 9420*n^6 - 216045*n^5 + 378090*n^4 + 3192694*n^3 - 6899520*n^2 - 19450080*n + 48327120), n>=7.

G.f.: x^5*(622*x^14 - 6966*x^13 + 37088*x^12 - 130876*x^11 + 344918*x^10 - 655255*x^9 + 737997*x^8 - 153262*x^7 - 639936*x^6 + 251910*x^5 + 1132096*x^4 - 1113158*x^3 - 443028*x^2 - 27305*x - 125)/(x-1)^13.

cp's OEIS Frontend

A194652 Number of ways to place 4 nonattacking kings on an n X n cylindrical chessboard.

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A194653 Number of ways to place 5 nonattacking kings on an n X n cylindrical chessboard.

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A194654 Number of ways to place 6 nonattacking kings on an n X n cylindrical chessboard.

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