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

1,1

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).

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)).

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 *)

Prog

(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

Crossrefs

Cf. A061593, A137432, A195590.

Sequence in context: A183682 A243027 A242543 * A068381 A143238 A102091

Adjacent sequences: A194641 A194642 A194643 * A194645 A194646 A194647

Keyword

nonn,easy

Author

Vaclav Kotesovec, Aug 31 2011

Status

approved