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 (3,0,-4).
FORMULA
G.f.: (1-3*x^2)/(1 - 3*x + 4*x^3).
a(n) = 3*a(n-1) - 4*a(n-3) with a(0)=1, a(1)=3 and a(2)=6.
a(n) = ((3*n+22)*2^n - 4*(-1)^n)/18.
MAPLE
with(LinearAlgebra): nmax:=29; m:=5; A[5]:= [0, 0, 0, 0, 0, 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 - 3 x^2)/(1 - 3 x + 4 x^3), {x, 0, 29}], x] (* Michael De Vlieger, Nov 02 2018 *)
LinearRecurrence[{3, 0, -4}, {1, 3, 6}, 30] (* Harvey P. Dale, Aug 12 2020 *)
PROG
(Magma) [((3*n+22)*2^n-4*(-1)^n)/18: n in [0..40]]; // Vincenzo Librandi, Aug 04 2011
(PARI) vector(40, n, n--; ((3*n+22)*2^n - 4*(-1)^n)/18) \\ G. C. Greubel, Nov 03 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Johannes W. Meijer, Aug 06 2010
STATUS
approved