A179603 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 3*x)/(1 - 3*x - 7*x^2).

1, 6, 25, 117, 526, 2397, 10873, 49398, 224305, 1018701, 4626238, 21009621, 95412529, 433304934, 1967802505, 8936542053, 40584243694, 184308525453, 837015282217, 3801205524822, 17262723549985, 78396609323709

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 king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596.

The sequence above corresponds to 24 red king vectors, i.e., A[5] vectors, with decimal values 95, 119, 125, 215, 221, 245, 287, 311, 317, 347, 350, 371, 374, 377, 380, 407, 413, 437, 467, 470, 473, 476, 497 and 500. These vectors lead for the corner squares to A015524 and for the side squares to A179602.

Links

Table of n, a(n) for n=0..21.

Index entries for linear recurrences with constant coefficients, signature (3, 7).

Formula

G.f.: (1+3*x)/(1 - 3*x - 7*x^2).

a(n) = 3*a(n-1) + 7*a(n-2) with a(0) = 1 and a(1) = 6.

a(n) = ((9+5*A)*A^(-n-1) + (9+5*B)*B^(-n-1))/37 with A = (-3+sqrt(37))/14 and B = (-3-sqrt(37))/14.

Maple

with(LinearAlgebra): nmax:=23; m:=5; A[1]:= [0, 1, 0, 1, 1, 0, 0, 0, 0]: A[2]:= [1, 0, 1, 1, 1, 1, 0, 0, 0]: A[3]:= [0, 1, 0, 0, 1, 1, 0, 0, 0]: A[4]:= [1, 1, 0, 0, 1, 0, 1, 1, 0]: A[5]:= [1, 1, 1, 0, 1, 0, 0, 1, 1]: A[6]:= [0, 1, 1, 0, 1, 0, 0, 1, 1]: A[7]:= [0, 0, 0, 1, 1, 0, 0, 1, 0]: A[8]:= [0, 0, 0, 1, 1, 1, 1, 0, 1]: A[9]:= [0, 0, 0, 0, 1, 1, 0, 1, 0]: A:=Matrix([A[1], A[2], A[3], A[4], A[5], A[6], A[7], A[8], A[9]]): 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. A179597 (central square).

Sequence in context: A346818 A120758 A227914 * A298700 A215763 A153481

Adjacent sequences: A179600 A179601 A179602 * A179604 A179605 A179606

Keyword

easy,nonn

Author

Johannes W. Meijer, Jul 28 2010

Status

approved