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

1,1

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

Table of n, a(n) for n=1..24.

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).

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)).

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 *)

Crossrefs

Cf. A061594, A194644, A137432, A195591.

Sequence in context: A188862 A228686 A172517 * A134845 A167982 A063498

Adjacent sequences: A194642 A194643 A194644 * A194646 A194647 A194648

Keyword

nonn

Author

Vaclav Kotesovec, Aug 31 2011

Status

approved