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A179603 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 3*x)/(1 - 3*x - 7*x^2).

%I A179603 #6 Jun 30 2023 16:24:05
%S A179603 1,6,25,117,526,2397,10873,49398,224305,1018701,4626238,21009621,
%T A179603 95412529,433304934,1967802505,8936542053,40584243694,184308525453,
%U A179603 837015282217,3801205524822,17262723549985,78396609323709
%N A179603 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 A179603 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 A179603 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 A179603 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3, 7).
%F A179603 G.f.: (1+3*x)/(1 - 3*x - 7*x^2).
%F A179603 a(n) = 3*a(n-1) + 7*a(n-2) with a(0) = 1 and a(1) = 6.
%F A179603 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 A179603 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 A179603 Cf. A179597 (central square).
%K A179603 easy,nonn
%O A179603 0,2
%A A179603 _Johannes W. Meijer_, Jul 28 2010