A179610 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: 1/(1-3*x-5*x^2+4*x^3).

1, 3, 14, 53, 217, 860, 3453, 13791, 55198, 220737, 883037, 3532004, 14128249, 56512619, 226051086, 904203357, 3616815025, 14467257516, 57869034245, 231476130215, 925904531806, 3703618109513, 14814472466709, 59257889820468

Offset: 0

Author

Johannes W. Meijer, Jul 28 2010, revised Aug 15 2010

Keywords

• easy
• nonn

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 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 4 red king vectors, i.e. A[5] vectors, with decimal [binary] values 85 [0,0,1,0,1,0,1,0,1], 277 [1,0,0,0,1,0,1,0,1], 337 [1,0,1,0,1,0,0,0,1] and 340 [1,0,1,0,1,0,1,0,0].

Convolution of (-4)^n and F(n+1) with F = A000045.

Links

• Index entries for linear recurrences with constant coefficients, signature (3, 5, -4).

Programs

Maple

with(LinearAlgebra): nmax:=23; m:=1; 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]:= [1,0,0,0,1,0,1,0,1]: 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);

Mathematica

CoefficientList[Series[1/(1-3x-5x^2+4x^3),{x,0,30}],x] (* or *) LinearRecurrence[{3,5,-4},{1,3,14},30] (* Harvey P. Dale, Aug 12 2025 *)

Formula

G.f.: = 1/((x^2-x-1)*(4*x-1)).

a(n) = 3*a(n-1)+5*a(n-2)-4*a(n-3) with a(1)=1, a(2)=3 and a(3)=14.

a(n) = (1/95)*(5*2^(2*n+4)-(11-2*phi)*phi^(-n-1)-(9+2*phi)*(1-phi)^(-n-1)) with phi = (1+sqrt(5))/2, with A001622 = phi.

a(n) = (-1)^n*sum((-4)^m*F(n+1-m),m=0..n).