A180147 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + 3*x)/(1 - 4*x - 3*x^2 + 6*x^3).

1, 7, 31, 139, 607, 2659, 11623, 50827, 222223, 971635, 4248247, 18574555, 81213151, 355086787, 1552539271, 6788138539, 29679651247, 129767784979, 567381262423, 2480750497147, 10846539065983, 47424120180835

Offset

0,2

Comments

The a(n) represent the number of n-move routes of a fairy chess piece starting in the central square (m = 5) 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.

On a 3 X 3 chessboard there are 2^9 = 512 ways to go berserk on the central square (we assume here that a berserker might behave like a rook). The berserker is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program. For the central squares the 512 berserkers lead to 42 berserker sequences, see the cross-references for some examples.

The sequence above corresponds to six A[5] vectors with decimal values between 191 and 506. These vectors lead for the corner squares to A180145 and for the side squares to A180146.

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

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

Formula

G.f.: (1+3*x)/(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)=7 and a(2)=31.

a(n) = -1/2 + (7+6*A)*A^(-n-1)/22 + (7+6*B)*B^(-n-1)/22 with A=(-3+sqrt(33))/12 and B=(-3-sqrt(33))/12.

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

Maple

with(LinearAlgebra): nmax:=22; m:=5; 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

CoefficientList[Series[(1+3x)/(1-4x-3x^2+6x^3), {x, 0, 40}], x] (* or *) LinearRecurrence[{4, 3, -6}, {1, 7, 31}, 40] (* Harvey P. Dale, Oct 10 2011 *)

Crossrefs

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

Cf. Berserker sequences central square [numerical values A[5]]: A000007 [0], A000012 [16], 2*A001835 [17, n>=1 and a(0)=1], A155116 [3], A077829 [7], A000302 [15], 6*A179606 [111, with leading 1 added], 2*A033887 [95, n>=1 and a(0)=1], A180147 [191, this sequence], 2*A180141 [495, n>=1 and a(0)=1], 4*A107979 [383, with leading 1 added].

Sequence in context: A262630 A282857 A296573 * A044049 A378687 A255284

Adjacent sequences: A180144 A180145 A180146 * A180148 A180149 A180150

Keyword

easy,nonn

Author

Johannes W. Meijer, Aug 13 2010

Status

approved