OFFSET
0,2
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).
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).
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);
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Johannes W. Meijer, Jul 28 2010, revised Aug 15 2010
STATUS
approved