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A179604 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 2*x - 9*x^2 - 2*x^3).

%I A179604 #15 Apr 07 2024 08:49:02
%S A179604 1,3,15,59,259,1079,4607,19443,82507,349215,1479879,6267707,26552755,
%T A179604 112474631,476459471,2018296131,8549676763,36216937647,153417558423,
%U A179604 649886909195,2752965719491,11661748738583,49399962770975
%N A179604 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 2*x - 9*x^2 - 2*x^3).
%C A179604 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 A179604 The sequence above corresponds to 4 red king vectors, i.e., 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 side squares to A015448 and for the central square to A179605.
%H A179604 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,9,2).
%F A179604 G.f.: ( -1-x ) / ( (2*x+1)*(x^2 + 4*x - 1) ).
%F A179604 a(n) = 2*a(n-1) + 9*a(n-2) + 2*a(n-3) with a(0)=1, a(1)=3 and a(2)=15.
%F A179604 a(n) = (20*(-1/2)^(-n) + (5+7*sqrt(5))*A^(-n-1) + (5-7*sqrt(5))*B^(-n-1))/110 with A = (-2+sqrt(5)) and B:= (-2-sqrt(5)).
%F A179604 Limit_{k->oo} a(n+k)/a(k) = (-1)^(n+1)/(A001076(n)*sqrt(5) - A001077(n)).
%p A179604 with(LinearAlgebra): nmax:=22; 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]:= [1,0,1,1,0,0,1,0,1]: 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);
%t A179604 LinearRecurrence[{2,9,2},{1,3,15},30] (* or *) CoefficientList[ Series[ (x+1)/(-2 x^3-9 x^2-2 x+1),{x,0,30}],x] (* _Harvey P. Dale_, Mar 17 2012 *)
%Y A179604 Cf. A001076, A001077, A015448, A179605, A179596.
%K A179604 easy,nonn
%O A179604 0,2
%A A179604 _Johannes W. Meijer_, Jul 28 2010