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A179607 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 2*x - 4*x^2)/(1 - 2*x - 8*x^2).

%I A179607 #11 Mar 08 2021 11:53:26
%S A179607 1,4,12,56,208,864,3392,13696,54528,218624,873472,3495936,13979648,
%T A179607 55926784,223690752,894795776,3579117568,14316601344,57266143232,
%U A179607 229065097216,916259340288,3665039458304,14660153638912,58640622944256
%N A179607 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 2*x - 4*x^2)/(1 - 2*x - 8*x^2).
%C A179607 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 A179607 The sequence above corresponds to just one red king vector, i.e., A[5] vector, with decimal [binary] value 325 [1,0,1,0,0,0,1,0,1]. This vectors leads for the corner squares to A083424 and for the side squares to A003947.
%C A179607 The inverse binomial transform of A100284 (without the first leading 1).
%H A179607 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2, 8).
%F A179607 G.f.: (1 + 2*x - 4*x^2)/(1 - 2*x - 8*x^2).
%F A179607 a(n) = 2*a(n-1) + 8*a(n-2), for n >= 3, with a(0) = 1, a(1) = 4 and a(2) = 12.
%F A179607 a(n) = 5*(4)^(n)/6 - (-2)^(n)/3 for n >= 1 and a(0) = 1.
%F A179607 a(n) = 4*A083424(n-1), n>0. - _R. J. Mathar_, Mar 08 2021
%p A179607 with(LinearAlgebra): nmax:=24; 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]:= [1,0,1,0,0,0,1,0,1]: 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 A179607 Join[{1},LinearRecurrence[{2,8},{4,12},30]] (* _Harvey P. Dale_, Mar 01 2012 *)
%Y A179607 Cf. A179597 (central square).
%K A179607 easy,nonn
%O A179607 0,2
%A A179607 _Johannes W. Meijer_, Jul 28 2010