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.
The sequence above corresponds to just one red king vector, i.e., A[5] vector, with decimal [binary] value 325 [1,0,1,0,0,0,1,0,1]. This vectors leads for the corner squares to A083424 and for the side squares to A003947.
The inverse binomial transform of A100284 (without the first leading 1).
LINKS
FORMULA
G.f.: (1 + 2*x - 4*x^2)/(1 - 2*x - 8*x^2).
a(n) = 2*a(n-1) + 8*a(n-2), for n >= 3, with a(0) = 1, a(1) = 4 and a(2) = 12.
a(n) = 5*(4)^(n)/6 - (-2)^(n)/3 for n >= 1 and a(0) = 1.
a(n) = 4*A083424(n-1), n>0. - R. J. Mathar, Mar 08 2021
MAPLE
with(LinearAlgebra): nmax:=24; 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, 0, 1, 0, 0, 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
Join[{1}, LinearRecurrence[{2, 8}, {4, 12}, 30]] (* Harvey P. Dale, Mar 01 2012 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Johannes W. Meijer, Jul 28 2010
STATUS
approved