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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).

%I A179599 #6 Jun 30 2023 16:22:23
%S A179599 1,7,29,143,661,3127,14669,69023,324421,1525447,7171709,33718703,
%T A179599 158529781,745338967,3504255149,16475477183,77460472741,364185235687,
%U A179599 1712239488989,8050200352463,37848516969301,177947153727607
%N 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).
%C A179599 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 A179599 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.
%H A179599 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3, 8).
%F A179599 G.f.: (1+4*x)/(1 - 3*x - 8*x^2).
%F A179599 a(n) = 3*a(n-1) + 8*a(n-2) with a(0) = 1 and a(1) = 7.
%F A179599 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.
%p A179599 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);
%Y A179599 Cf. A179597 (central square).
%K A179599 easy,nonn
%O A179599 0,2
%A A179599 _Johannes W. Meijer_, Jul 28 2010