A194645 Number of ways to place 3n nonattacking kings on a 6 X 2n cylindrical chessboard.

32, 100, 344, 1220, 4392, 15988, 58776, 218052, 815816, 3076180, 11682296, 44653028, 171670440, 663421684, 2575592664, 10039703172, 39273896840, 154109956756, 606353229752, 2391296071460, 9449664931176, 37407140524084, 148300497571992, 588693691298244

Offset: 1

Author

Vaclav Kotesovec, Aug 31 2011

Keywords

• nonn

Comments

This cylinder is horizontal: a chessboard where it is supposed that rows 1 and 2n are in contact (number of columns = 6, number of rows = 2n).

Links

• Ray Chandler, Table of n, a(n) for n = 1..1660
• V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights
• Index entries for linear recurrences with constant coefficients, signature (12,-53,104,-86,24).

Crossrefs

Cf. A061594, A194644, A137432, A195591.

Programs

Mathematica

Table[FullSimplify[2*4^n+2*3^n+4*(2+Sqrt[2])^n+4*(2-Sqrt[2])^n+2], {n,25}]
LinearRecurrence[{12,-53,104,-86,24},{32,100,344,1220,4392},30] (* Harvey P. Dale, Jul 25 2016 *)

Formula

a(n) = 2*4^n + 2*3^n + 4*(2+sqrt(2))^n + 4*(2-sqrt(2))^n + 2.

Recurrence: a(n) = 24*a(n-5) - 86*a(n-4) + 104*a(n-3) - 53*a(n-2) + 12*a(n-1).

G.f.: -2*(7-68*x+229*x^2-308*x^3+134*x^4)/((-1+x)*(-1+3*x)*(-1+4*x)*(1-4*x+2*x^2)).