A194644 Number of ways to place 2n nonattacking kings on a 4 X 2n cylindrical chessboard.

12, 32, 90, 256, 732, 2102, 6060, 17536, 50922, 148352, 433500, 1270246, 3731532, 10987232, 32418810, 95835136, 283784412, 841611542, 2499330540, 7431221056, 22118855562, 65898914432, 196498594140, 586358987206, 1750864725132, 5231094261152, 15636995277210

Offset: 1

Author

Vaclav Kotesovec, Aug 31 2011

Keywords

• nonn
• easy

Comments

This cylinder is horizontal: a chessboard where it is supposed that rows 1 and 2n are in contact (number of columns = 4, number of rows = 2n).

Links

• Indranil Ghosh, Table of n, a(n) for n = 1..2091
• V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights
• Index entries for linear recurrences with constant coefficients, signature (6,-10,3).

Crossrefs

Cf. A061593, A137432, A195590.

Programs

Mathematica

Table[2*3^n+2*LucasL[2n], {n,25}]
Drop[CoefficientList[Series[-2*(3 - 12*x + 10*x^2)/((-1 + 3*x)*(1 - 3*x + x^2)), {x, 0, 27}],x], 1] (* or *) LinearRecurrence[{6,-10,3},{12,32,90},27] (* Indranil Ghosh, Mar 05 2017 *)

PARI

print(Vec(-2*(3 - 12*x + 10*x^2)/((-1 + 3*x)*(1 - 3*x + x^2)) + O(x^27))); \\ Indranil Ghosh, Mar 05 2017

Formula

a(n) = 2*3^n + 2*((3+sqrt(5))/2)^n + 2*((3-sqrt(5))/2)^n.

Recurrence: a(n) = 3*a(n-3) - 10*a(n-2) + 6*a(n-1).

G.f.: -2*(3-12*x+10*x^2)/((-1+3*x)*(1-3*x+x^2)).