A180145 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 - 3*x^2)/(1 - 4*x - 3*x^2 + 6*x^3).

1, 4, 16, 70, 304, 1330, 5812, 25414, 111112, 485818, 2124124, 9287278, 40606576, 177543394, 776269636, 3394069270, 14839825624, 64883892490, 283690631212, 1240375248574, 5423269532992, 23712060090418, 103675797469204

Offset

0,2

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 6 A[5] vectors with decimal values between 191 and 506. These vectors lead for the side squares to A180146 and for the central square to A180147.

Links

Table of n, a(n) for n=0..22.

Index entries for linear recurrences with constant coefficients, signature (4, 3, -6).

Formula

G.f.: (1-3*x^2)/(1 - 4*x - 3*x^2 + 6*x^3).

a(n) = 4*a(n-1) + 3*a(n-2) - 6*a(n-3) with a(0)=1, a(1)=4 and a(2)=16.

a(n) = 1/4 + (7+6*A)*A^(-n-1)/44 + (7+6*B)*B^(-n-1)/44 with A=(-3+sqrt(33))/12 and B=(-3-sqrt(33))/12.

a(n) = A180146(n) - 3*A180146(n-2) with A180146(-2) = A180146(-1) = 0.

Maple

with(LinearAlgebra): nmax:=22; m:=1; A[5]:=[0, 1, 0, 1, 1, 1, 1, 1, 1]: 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);

Crossrefs

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

Sequence in context: A298048 A344267 A144316 * A133789 A151244 A280767

Adjacent sequences: A180142 A180143 A180144 * A180146 A180147 A180148

Keyword

easy,nonn

Author

Johannes W. Meijer, Aug 13 2010

Status

approved