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%I #6 Jun 30 2023 16:24:05
%S 1,6,25,117,526,2397,10873,49398,224305,1018701,4626238,21009621,
%T 95412529,433304934,1967802505,8936542053,40584243694,184308525453,
%U 837015282217,3801205524822,17262723549985,78396609323709
%N Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 3*x)/(1 - 3*x - 7*x^2).
%C 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.
%C The sequence above corresponds to 24 red king vectors, i.e., A[5] vectors, with decimal values 95, 119, 125, 215, 221, 245, 287, 311, 317, 347, 350, 371, 374, 377, 380, 407, 413, 437, 467, 470, 473, 476, 497 and 500. These vectors lead for the corner squares to A015524 and for the side squares to A179602.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3, 7).
%F G.f.: (1+3*x)/(1 - 3*x - 7*x^2).
%F a(n) = 3*a(n-1) + 7*a(n-2) with a(0) = 1 and a(1) = 6.
%F a(n) = ((9+5*A)*A^(-n-1) + (9+5*B)*B^(-n-1))/37 with A = (-3+sqrt(37))/14 and B = (-3-sqrt(37))/14.
%p with(LinearAlgebra): nmax:=23; 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]:= [1,1,1,0,1,0,0,1,1]: 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);
%Y Cf. A179597 (central square).
%K easy,nonn
%O 0,2
%A _Johannes W. Meijer_, Jul 28 2010
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