OFFSET
0,2
COMMENTS
The a(n) represent the number of n-move routes of a fairy chess piece starting in a given side square (m = 2, 4, 6 or 8) 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 10 red king vectors, i.e., A[5] vectors, with decimal values 239, 351, 375, 381, 431, 471, 477, 491, 494, and 501. These vectors lead for the corner squares to A015525 and for the central square to A179599.
Inverse binomial transform of A126501.
LINKS
FORMULA
G.f.: (1+2*x)/(1 - 3*x - 8*x^2).
a(n) = 3*a(n-1) + 8*a(n-2) with a(0) = 1 and a(1) = 5.
a(n) = ((41+5*sqrt(41))*A^(-n-1) + (41-5*sqrt(41))*B^(-n-1))/328 with A = (-3+sqrt(41))/16 and B = (-3-sqrt(41))/16.
MAPLE
with(LinearAlgebra): nmax:=21; m:=2; 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, 1, 1, 1, 1, 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
STATUS
approved