A194650 -id:A194650 - cp's OEIS frontend

A212269 Number of ways to place k non-attacking kings on an n X n cylindrical chessboard, summed over all k >= 0.

2, 5, 19, 205, 3011, 92875, 4763459, 459630701, 78223965193, 24270274906085, 13497818986883771, 13571363009654254429, 24562890586806439035377, 80199120146273882569630015, 471874707649862024071657639861, 5005895207027974222377733802848093

Offset: 1

Vaclav Kotesovec, May 12 2012

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 343.

Vaclav Kotesovec, Table of n, a(n) for n = 1..27
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p.165

Cf. A063443 (n+1), A067958, A194650-A194654, A182407, A247413, A212269.

Limit n ->infinity (a(n))^(1/n^2) = 1.342643951124... (see A247413).

A194651 Number of ways to place 3 nonattacking kings on an n X n cylindrical chessboard.

Original entry on oeis.org

0, 0, 0, 88, 785, 3528, 11151, 28560, 63513, 127520, 236863, 413736, 687505, 1096088, 1687455, 2521248, 3670521, 5223600, 7286063, 9982840, 13460433, 17889256, 23466095, 30416688, 38998425, 49503168, 62260191, 77639240, 96053713, 117963960, 143880703

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 (7, -21, 35, -35, 21, -7, 1).

Cf. A061996, A179404, A194650.

CoefficientList[Series[-x^3*(15*x^6 - 89*x^5 + 196*x^4 - 140*x^3 - 119*x^2 + 169*x + 88)/(x - 1)^7, {x, 0, 30}], x] (* Wesley Ivan Hurt, Dec 27 2023 *)

a(n) = 1/6*n*(n^5 - 27*n^3 + 18*n^2 + 194*n - 228), n>=4.

G.f.: -x^4*(15*x^6 - 89*x^5 + 196*x^4 - 140*x^3 - 119*x^2 + 169*x + 88)/(x-1)^7.

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

Original entry on oeis.org

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

A212269 Number of ways to place k non-attacking kings on an n X n cylindrical chessboard, summed over all k >= 0.

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A194651 Number of ways to place 3 nonattacking kings on an n X n cylindrical chessboard.

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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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