A180146 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: 1/(1 - 4*x - 3*x^2 + 6*x^3).

1, 4, 19, 82, 361, 1576, 6895, 30142, 131797, 576244, 2519515, 11016010, 48165121, 210591424, 920764999, 4025843542, 17602120621, 76961423116, 336496993075, 1471259517922, 6432760512217, 28125838644184, 122974079005855

Offset

0,2

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 6 A[5] vectors with decimal values between 191 and 506. These vectors lead for the corner squares to A180145 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/(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(-2)=0, a(-1)=0, a(0)=1, a(1)=4 and a(2)=19.

a(n) = (-1/8) + (13+30*A)*A^(-n-1)/88 + (13+30*B)*B^(-n-1)/88 with A=(-3+sqrt(33))/12 and B=(-3-sqrt(33))/12.

Maple

with(LinearAlgebra): nmax:=22; m:=2; 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);

Mathematica

Join[{a=1, b=4}, Table[c=3*b+6*a+1; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 18 2011 *)

Crossrefs

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

Sequence in context: A326649 A050914 A017961 * A017962 A260746 A290667

Adjacent sequences: A180143 A180144 A180145 * A180147 A180148 A180149

Keyword

easy,nonn

Author

Johannes W. Meijer, Aug 13 2010

Status

approved