A195594 Number of ways to place 6n nonattacking kings on a vertical cylinder 12 X 2n.

128, 2102, 15988, 81606, 327192, 1118398, 3419648, 9643562, 25603228, 64923594, 158877948, 378088270, 879980720, 2011806182, 4532900488, 10091643138, 22244251284, 48622120786, 105526014500, 227633451206, 488451508168, 1043298475662, 2219419264848

Offset: 1

Author

Vaclav Kotesovec, Sep 21 2011

Keywords

• nonn

Comments

Vertical cylinder: a chessboard where it is supposed that the columns 1 and 12 are in contact (number of columns = 12, number of rows = 2n).

Links

• Ray Chandler, Table of n, a(n) for n = 1..3296
• V. Kotesovec, Non-attacking chess pieces
• Index entries for linear recurrences with constant coefficients, signature (9, -34, 70, -85, 61, -24, 4).

Crossrefs

Cf. A194648, A174154, A137432.

Programs

Mathematica

LinearRecurrence[{9,-34,70,-85,61,-24,4},{128,2102,15988,81606,327192,1118398,3419648},30] (* Harvey P. Dale, Aug 06 2024 *)

Formula

Recurrence: a(n) = 4*a(n-7) - 24*a(n-6) + 61*a(n-5) - 85*a(n-4) + 70*a(n-3) - 34*a(n-2) + 9*a(n-1).

G.f.: -(1 + 119*x + 984*x^2 + 1352*x^3 + 307*x^4 + 9*x^5)/((x-1)^5*(2*x-1)^2).

a(n) = (15839*n - 99729)*2^n + 231/2*n^4 + 1767*n^3 + 26001/2*n^2 + 53295*n + 99730.