%I #7 Mar 10 2020 21:54:27
%S 1,3,16,66,304,1332,5968,26472,117952,524496,2334400,10385568,
%T 46213120,205619520,914912512,4070872704,18113348608,80595074304,
%U 358607125504,1595618388480,7099688329216,31589989045248,140559334936576
%N Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 2*x - 10*x^2 - 4*x^3).
%C 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 The sequence above corresponds to 6 red king vectors, i.e., A[5] vectors, with decimal values 335, 359, 365, 455, 461 and 485. These vectors lead for the side squares to A123347 and for the central square to A179601.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,10,4).
%F G.f.: (1+x)/(1 - 2*x - 10*x^2 - 4*x^3).
%F a(n) = 2*a(n-1) + 10*a(n-2) + 4*a(n-3) with a(0)=1, a(1)=3 and a(2)=16.
%F a(n) = (4*(-1/2)^(-n) + (1+sqrt(6))*A^(-n-1) + (1-sqrt(6))*B^(-n-1))/20 with A = (-1+sqrt(6)/2) and B = (-1-sqrt(6)/2).
%F Lim_{k->infinity} a(n+k)/a(k) = (-1)^(n+1)*(A016116(n+1)/(A041007(n-1)*sqrt(6) - A041006(n-1))) for n => 1.
%p with(LinearAlgebra): nmax:=24; 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]:= [1,1,1,0,0,0,1,1,1]: 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);
%o (PARI) Vec((1+x)/(1 - 2*x - 10*x^2 - 4*x^3) + O(x^40)) \\ _Jinyuan Wang_, Mar 10 2020
%K nonn,easy
%O 0,2
%A _Johannes W. Meijer_, Jul 28 2010