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A179610
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Eight white kings and one red king on a 3 X 3 chessboard. G.f.: 1/(1-3*x-5*x^2+4*x^3).
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3
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1, 3, 14, 53, 217, 860, 3453, 13791, 55198, 220737, 883037, 3532004, 14128249, 56512619, 226051086, 904203357, 3616815025, 14467257516, 57869034245, 231476130215, 925904531806, 3703618109513, 14814472466709, 59257889820468
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OFFSET
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0,2
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COMMENTS
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The a(n) represent the number of n-move routes of a fairy chess piece starting in a given corner square (m = 1, 3, 7 or 9) 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 4 red king vectors, i.e. A[5] vectors, with decimal [binary] values 85 [0,0,1,0,1,0,1,0,1], 277 [1,0,0,0,1,0,1,0,1], 337 [1,0,1,0,1,0,0,0,1] and 340 [1,0,1,0,1,0,1,0,0].
Convolution of (-4)^n and F(n+1) with F = A000045.
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LINKS
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FORMULA
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G.f.: = 1/((x^2-x-1)*(4*x-1)).
a(n) = 3*a(n-1)+5*a(n-2)-4*a(n-3) with a(1)=1, a(2)=3 and a(3)=14.
a(n) = (1/95)*(5*2^(2*n+4)-(11-2*phi)*phi^(-n-1)-(9+2*phi)*(1-phi)^(-n-1)) with phi = (1+sqrt(5))/2, with A001622 = phi.
a(n) = (-1)^n*sum((-4)^m*F(n+1-m),m=0..n).
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MAPLE
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with(LinearAlgebra): nmax:=23; m:=1; 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, 0, 0, 1, 0, 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);
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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