A180034 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1 - 2*x)/(1 - 6*x + 2*x^2).

1, 4, 22, 124, 700, 3952, 22312, 125968, 711184, 4015168, 22668640, 127981504, 722551744, 4079347456, 23030981248, 130027192576, 734101192960, 4144552772608, 23399114249728, 132105579953152, 745835251219456

Offset: 0

Author

Johannes W. Meijer, Aug 09 2010

Keywords

• easy
• nonn

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 white queen on the eight side and corner squares but on the central square the queen explodes with fury and turns into a red queen, see A180028.

The sequence above corresponds to 56 red queen vectors, i.e., A[5] vector, with decimal values varying between 23 and 464. The corner and side squares lead for these vectors to A154244.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (6,-2).

Crossrefs

Cf. A154244, A180028.

Programs

Magma

I:=[1,4]; [n le 2 select I[n] else 6*Self(n-1)-2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 15 2011

Maple

with(LinearAlgebra): nmax:=21; m:=5; A[5]:= [0,0,0,0,1,0,1,1,1]: A:=Matrix([[0,1,1,1,1,0,1,0,1], [1,0,1,1,1,1,0,1,0], [1,1,0,0,1,1,1,0,1], [1,1,0,0,1,1,1,1,0], A[5], [0,1,1,1,1,0,0,1,1], [1,0,1,1,1,0,0,1,1], [0,1,0,1,1,1,1,0,1], [1,0,1,0,1,1,1,1,0]]): 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

LinearRecurrence[{6,-2},{1,4},50] (* Vincenzo Librandi, Nov 15 2011 *)

Formula

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

a(n) = 6*a(n-1) - 2*a(n-2) with a(0) = 1 and a(1) = 4.

a(n) = ((1+4*A)*A^(-n-1) + (1+4*B)*B^(-n-1))/14 with A = (3+sqrt(7))/2 and B = (3-sqrt(7))/2.

a(n) = A154244(n) - 2*A154244(n-1). - R. J. Mathar, Aug 14 2012