A194646 Number of ways to place 4n nonattacking kings on an 8 X 2n cylindrical chessboard.

80, 276, 1082, 4460, 18890, 81606, 358564, 1599820, 7238864, 33175486, 153802520, 720390254, 3404944506, 16221905696, 77820675992, 375564803020, 1821845982082, 8876847931644, 43416046650306, 213033152875350, 1048198981050148, 5169676077206180

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 = 8, number of rows = 2n).

Links

Ray Chandler, Table of n, a(n) for n = 1..1430

V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights

Index entries for linear recurrences with constant coefficients, signature (29, -369, 2708, -12676, 39539, -83449, 118727, -111545, 66422, -23320, 4235, -300).

Formula

a(n) = 4 + 2*5^n + 2*4^n + 2*(2+sqrt(3))^n+2*(2-sqrt(3))^n + 4*((5+sqrt(5))/2)^n+4*((5-sqrt(5))/2)^n + 4*((5+sqrt(13))/2)^n+4*((5-sqrt(13))/2)^n + 2*(2*cos(Pi/7))^(2n)+2*(2*cos(2*Pi/7))^(2n)+2*(2*cos(3*Pi/7))^(2n).

Recurrence: a(n) = -300*a(n-12) + 4235*a(n-11) - 23320*a(n-10) + 66422*a(n-9) - 111545*a(n-8) + 118727*a(n-7) - 83449*a(n-6) + 39539*a(n-5) - 12676*a(n-4) + 2708*a(n-3) - 369*a(n-2) + 29*a(n-1).

G.f.: -2*(-17 + 453*x - 5251*x^2 + 34737*x^3 - 144635*x^4 + 394423*x^5 - 711101*x^6 + 836705*x^7 - 620007*x^8 + 270365*x^9 - 61055*x^10 + 5335*x^11)/((-1+x)*(-1+4*x)*(-1+5*x)*(1-4*x+x^2)*(1-5*x+3*x^2)*(1-5*x+5*x^2)*(-1+5*x-6*x^2+x^3)).

Mathematica

Table[FullSimplify[4+2*5^n+2*4^n + 2*(2+Sqrt[3])^n + 2*(2-Sqrt[3])^n+ 4*((5+Sqrt[5])/2)^n + 4*((5-Sqrt[5])/2)^n + 4*((5+Sqrt[13])/2)^n+4*((5-Sqrt[13])/2)^n+ 2*(2*Cos[Pi/7])^(2n) + 2*(2*Cos[2*Pi/7])^(2n) + 2*(2*Cos[3*Pi/7])^(2n)], {n, 10}]

Crossrefs

Cf. A173782, A194644, A194645, A137432, A195592.

Sequence in context: A234890 A157912 A234883 * A057441 A204483 A204475

Adjacent sequences: A194643 A194644 A194645 * A194647 A194648 A194649

Keyword

nonn

Author

Vaclav Kotesovec, Aug 31 2011

Status

approved