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.
For the central square the 512 red kings lead to 47 different red king sequences, see the cross-references for some examples.
The sequence above corresponds to four A[5] vectors with decimal [binary] values 367 [1,0,1,1,0,1,1,1,1], 463 [1,1,1,0,0,1,1,1,1], 487 [1,1,1,1,0,0,1,1,1] and 493 [1,1,1,1,0,1,1,0,1]. These vectors lead for the corner squares to A179596 and for the side squares to A126473.
This sequence belongs to a family of sequences with g.f. (1 + (k+2)*x + (2*k-4)*x^2)/(1 - 2*x - (k+8)*x^2 - (2*k)*x^3). Red king sequences that are members of this family are A179607 (k=0), A179605 (k=1), A179601 (k=2), A179597 (k=3; this sequence) and A086348 (k=4). Another member of this family is A179609 (k = -4).
LINKS
FORMULA
G.f.: (1 + 5*x + 2*x^2)/(1 - 2*x - 11*x^2 - 6*x^3).
a(n) = 2*a(n-1) + 11*a(n-2) + 6*a(n-3) with a(0) = 1, a(1) = 7 and a(2) = 27.
a(n) = 8*(-1/2)^(-n+1)/9 + ((7+11*sqrt(7))*A^(-n-1) + (7-11*sqrt(7))*B^(-n-1))/126 with A = (-2+sqrt(7))/3 and B = (-2-sqrt(7))/3.
MAPLE
with(LinearAlgebra): nmax:=21; 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, 1, 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, 11, 6}, {1, 7, 27}, 30] (* Harvey P. Dale, Mar 01 2015 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Johannes W. Meijer, Jul 28 2010, Aug 10 2010
STATUS
approved