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 36 red king vectors, i.e., A[5] vectors, with decimal values 15, 39, 45, 75, 78, 99, 102, 105, 108, 135, 141, 165, 195, 198, 201, 204, 225, 228, 267, 270, 291, 294, 297, 300, 330, 354, 360, 387, 390, 393, 396, 417, 420, 450, 456 and 480.
LINKS
FORMULA
G.f.: (1+2*x)/(1 - 2*x - 8*x^2 - 4*x^3).
a(n) = 2*a(n-1) + 8*a(n-2) + 4*a(n-3) with a(1)=1, a(2)=4 and a(3)=16.
a(n) = (8 + 3*z1 - 6*z1^2)*z1^(-n)/(z1*37) + (8 + 3*z2 - 6*z2^2)*z2^(-n)/(z2*37) + (8 + 3*z3 - 6*z3^2)*z3^(-n)/(z3*37) with z1, z2 and z3 the roots of f(x) = 1 - 2*x - 8*x^2 - 4*x^3 = 0.
alpha = arctan(3*sqrt(111));
z1 = sqrt(10)*cos(alpha/3)/6 + sqrt(30)*sin(alpha/3)/6 - 2/3 = 0.2405971520460078;
z2 = -sqrt(10)*cos(alpha/3)/3 - 2/3 = -1.585043243313016;
z3 = sqrt(10)*cos(alpha/3)/6 - sqrt(30)*sin(alpha/3)/6 - 2/3 = -0.6555539087329909.
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]:= [0, 0, 0, 0, 0, 1, 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);
MATHEMATICA
LinearRecurrence[{2, 8, 4}, {1, 4, 16}, 30] (* Harvey P. Dale, Oct 20 2017 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Johannes W. Meijer, Jul 28 2010
STATUS
approved