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A179599
Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 4*x)/(1 - 3*x - 8*x^2).
3
1, 7, 29, 143, 661, 3127, 14669, 69023, 324421, 1525447, 7171709, 33718703, 158529781, 745338967, 3504255149, 16475477183, 77460472741, 364185235687, 1712239488989, 8050200352463, 37848516969301, 177947153727607
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 10 red king vectors, i.e., A[5] vectors, with decimal values 239, 351, 375, 381, 431, 471, 477, 491, 494 and 501. These vectors lead for the corner squares to A015525 and for the side squares to A179598.
FORMULA
G.f.: (1+4*x)/(1 - 3*x - 8*x^2).
a(n) = 3*a(n-1) + 8*a(n-2) with a(0) = 1 and a(1) = 7.
a(n) = ((11+4*A)*A^(-n-1) + (11+4*B)*B^(-n-1))/41 with A = (-3+sqrt(41))/16 and B = (-3-sqrt(41))/16.
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]:= [1, 0, 1, 1, 1, 1, 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);
CROSSREFS
Cf. A179597 (central square).
Sequence in context: A074468 A303091 A333887 * A266473 A297677 A287860
KEYWORD
easy,nonn
AUTHOR
Johannes W. Meijer, Jul 28 2010
STATUS
approved