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 bishop on the eight side and corner squares but on the central square the bishop turns into a raging elephant, see A175654.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-1,-6).
FORMULA
G.f.: (1+2*x-7*x^2)/(1-4*x+x^2+6*x^3).
a(n) = 4*a(n-1)-a(n-2)-6*a(n-3) with a(0)=1, a(1)=6 and a(2)=16.
a(n) = (-2*(-1)^n)/3-2^n/3+2*3^n. [Colin Barker, Oct 07 2012]
MAPLE
nmax:=24; m:=5; A[5]:= [1, 0, 1, 0, 1, 0, 1, 1, 1]: A:=Matrix([[0, 0, 0, 0, 1, 0, 0, 0, 1], [0, 0, 0, 1, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0, 1, 0, 0], [0, 1, 0, 0, 0, 0, 0, 1, 0], A[5], [0, 1, 0, 0, 0, 0, 0, 1, 0], [0, 0, 1, 0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 1, 0, 0, 0], [1, 0, 0, 0, 1, 0, 0, 0, 0]]): 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 + 2 x - 7 x^2) / (1 - 4 x + x^2 + 6 x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 21 2013 *)
PROG
(Magma) I:=[1, 6, 16]; [n le 3 select I[n] else 4*Self(n-1)-Self(n-2)-6*Self(n-3): n in [1..35]]; // Vincenzo Librandi, Jul 21 2013
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Johannes W. Meijer, Aug 06 2010
STATUS
approved