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.
LINKS
FORMULA
G.f.: ( -1-4*x ) / ( (2*x+1)*(2*x^2 + 4*x - 1) ).
a(n) = 2*a(n-1) + 10*a(n-2) + 4*a(n-3) with a(0)=1, a(1)=6 and a(2)=22.
a(n) = (-2/5)*(-1/2)^(-n) + ((2+3*A)*A^(-n-1) + (2+3*B)*B^(-n-1))/10 with A = (-1+sqrt(6)/2) and B = (-1-sqrt(6)/2).
MAPLE
with(LinearAlgebra): nmax:=22; 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, 0, 0, 1, 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
KEYWORD
easy,nonn
AUTHOR
Johannes W. Meijer, Jul 28 2010
STATUS
approved