A180143 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + x^2)/(1 - 4*x + x^2 + 2*x^3).

1, 4, 16, 58, 208, 742, 2644, 9418, 33544, 119470, 425500, 1515442, 5397328, 19222870, 68463268, 243835546, 868433176, 3092970622, 11015778220, 39233275906, 139731384160, 497660704294, 1772444881204, 6312656052202

Offset: 0

Author

Johannes W. Meijer, Aug 13 2010

Keywords

• easy
• nonn

Comments

The a(n) represent the number of n-move routes of a fairy chess piece starting in a given corner square (m = 1, 3, 7 or 9) on a 3 X 3 chessboard. This fairy chess piece behaves like a rook on the eight side and corner squares but on the central square the rook goes berserk and turns into a berserker, see A180140.

The sequence above corresponds to just one A[5] vectors with decimal value 16. This vector leads for the side squares to A180144 and for the central square to A000012.

Links

• Index entries for linear recurrences with constant coefficients, signature (4, -1, -2).

Crossrefs

Cf. A180141 (corner squares), A180140 (side squares), A180147 (central square).

Programs

Maple

with(LinearAlgebra): nmax:=23; m:=1; A[5]:=[0,0,0,0,1,0,0,0,0]: A:= Matrix([[0,1,1,1,0,0,1,0,0], [1,0,1,0,1,0,0,1,0], [1,1,0,0,0,1,0,0,1], [1,0,0,0,1,1,1,0,0], A[5], [0,0,1,1,1,0,0,0,1], [1,0,0,1,0,0,0,1,1], [0,1,0,0,1,0,1,0,1], [0,0,1,0,0,1,1,1,0]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);

Formula

G.f.: (1+x^2)/(1 - 4*x + x^2 + 2*x^3).

a(n) = 4*a(n-1) - 1*a(n-2) - 2*a(n-3) with a(0)=1, a(1)=4 and a(2)=16.

a(n) = -1/2 + (9+12*A)*A^(-n-1)/34 + (9+12*B)*B^(-n-1)/34 with A=(-3+sqrt(17))/4 and B=(-3-sqrt(17))/4.

Lim_{k->infinity} a(n+k)/a(k) = (-1)^(n)*(2)^(n+1)/((2*A007482(n) - 3*A007482(n-1)) - A007482(n-1)*sqrt(17)) for n >= 1.