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A179610 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).

%I A179610 #13 Aug 12 2025 16:23:41
%S A179610 1,3,14,53,217,860,3453,13791,55198,220737,883037,3532004,14128249,
%T A179610 56512619,226051086,904203357,3616815025,14467257516,57869034245,
%U A179610 231476130215,925904531806,3703618109513,14814472466709,59257889820468
%N A179610 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).
%C A179610 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.
%C A179610 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].
%C A179610 Convolution of (-4)^n and F(n+1) with F = A000045.
%H A179610 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3, 5, -4).
%F A179610 G.f.: = 1/((x^2-x-1)*(4*x-1)).
%F A179610 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.
%F A179610 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.
%F A179610 a(n) = (-1)^n*sum((-4)^m*F(n+1-m),m=0..n).
%p A179610 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);
%t A179610 CoefficientList[Series[1/(1-3x-5x^2+4x^3),{x,0,30}],x] (* or *) LinearRecurrence[{3,5,-4},{1,3,14},30] (* _Harvey P. Dale_, Aug 12 2025 *)
%K A179610 easy,nonn
%O A179610 0,2
%A A179610 _Johannes W. Meijer_, Jul 28 2010, revised Aug 15 2010