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A194645 Number of ways to place 3n nonattacking kings on a 6 X 2n cylindrical chessboard. 5
32, 100, 344, 1220, 4392, 15988, 58776, 218052, 815816, 3076180, 11682296, 44653028, 171670440, 663421684, 2575592664, 10039703172, 39273896840, 154109956756, 606353229752, 2391296071460, 9449664931176, 37407140524084, 148300497571992, 588693691298244 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 24 15:35 EST 2022. Contains 350538 sequences. (Running on oeis4.)