A180144 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 - 2*x^2)/(1 - 4*x + x^2 + 2*x^3).

1, 4, 13, 46, 163, 580, 2065, 7354, 26191, 93280, 332221, 1183222, 4214107, 15008764, 53454505, 190381042, 678052135, 2414918488, 8600859733, 30632416174, 109098967987, 388561736308, 1383883144897, 4928772907306

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 side square (m = 2, 4, 6 or 8) 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] vector with decimal value 16. This vector leads for the corner squares to A180143 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:=2; 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.: (1-2*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)=13.

a(n) = 1/4 + (21-6*A)*A^(-n-1)/68 + (21-6*B)*B^(-n-1)/68 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.