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A179605 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 3*x - 2*x^2)/(1 - 2*x - 9*x^2 - 2*x^3).

%I A179605 #12 Apr 07 2024 08:49:08
%S A179605 1,5,17,81,325,1413,5913,25193,106429,451421,1911089,8097825,34298293,
%T A179605 145299189,615478665,2607246617,11044399597,46784976077,198184041761,
%U A179605 839521667409,3556269662821,15064602415845,63814675131897
%N A179605 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 3*x - 2*x^2)/(1 - 2*x - 9*x^2 - 2*x^3).
%C A179605 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 toes crazy and turns into a red king, see A179596.
%C A179605 The sequence above corresponds to 4 red king vectors, A[5] vectors, with decimal [binary] values 327 [1,0,1,0,0,0,1,1,1], 333 [1,0,1,0,0,1,1,0,1], 357 [1,0,1,1,0,0,1,0,1] and 453 [1,1,1,0,0,0,1,0,1]. These vectors lead for the corner squares to A179604 and for the side squares to A015448.
%H A179605 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,9,2).
%F A179605 G.f.: ( -1 - 3*x + 2*x^2 ) / ( (2*x+1)*(x^2 + 4*x - 1) ).
%F A179605 a(n) = 2*a(n-1) + 9*a(n-2) + 2*a(n-3) with a(0)=1, a(1)=5 and a(2)=17.
%F A179605 a(n) = (-4/11)*(-1/2)^(-n) + ((17+41*A)*A^(-n-1) + (17+41*B)*B^(-n-1))/110 with A = (-2+sqrt(5)) and B =(-2-sqrt(5)).
%F A179605 Limit_{k->oo} a(n+k)/a(k) = (-1)^(n+1)/(A001076(n)*sqrt(5) - A001077(n)).
%p A179605 with(LinearAlgebra): nmax:=21; 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]:= [1,0,1,1,0,0,1,0,1]: 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);
%Y A179605 Cf. A001076, A001077, A015448, A179596, A179597 (central square), A179604.
%K A179605 easy,nonn
%O A179605 0,2
%A A179605 _Johannes W. Meijer_, Jul 28 2010