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%I #18 Apr 07 2024 09:09:11
%S 1,4,22,124,700,3952,22312,125968,711184,4015168,22668640,127981504,
%T 722551744,4079347456,23030981248,130027192576,734101192960,
%U 4144552772608,23399114249728,132105579953152,745835251219456
%N Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1 - 2*x)/(1 - 6*x + 2*x^2).
%C 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 white queen on the eight side and corner squares but on the central square the queen explodes with fury and turns into a red queen, see A180028.
%C The sequence above corresponds to 56 red queen vectors, i.e., A[5] vector, with decimal values varying between 23 and 464. The corner and side squares lead for these vectors to A154244.
%H Vincenzo Librandi, <a href="/A180034/b180034.txt">Table of n, a(n) for n = 0..200</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-2).
%F G.f.: (1-2*x)/(1 - 6*x + 2*x^2).
%F a(n) = 6*a(n-1) - 2*a(n-2) with a(0) = 1 and a(1) = 4.
%F a(n) = ((1+4*A)*A^(-n-1) + (1+4*B)*B^(-n-1))/14 with A = (3+sqrt(7))/2 and B = (3-sqrt(7))/2.
%F a(n) = A154244(n) - 2*A154244(n-1). - _R. J. Mathar_, Aug 14 2012
%p with(LinearAlgebra): nmax:=21; m:=5; A[5]:= [0,0,0,0,1,0,1,1,1]: A:=Matrix([[0,1,1,1,1,0,1,0,1], [1,0,1,1,1,1,0,1,0], [1,1,0,0,1,1,1,0,1], [1,1,0,0,1,1,1,1,0], A[5], [0,1,1,1,1,0,0,1,1], [1,0,1,1,1,0,0,1,1], [0,1,0,1,1,1,1,0,1], [1,0,1,0,1,1,1,1,0]]): 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 LinearRecurrence[{6,-2},{1,4},50] (* _Vincenzo Librandi_, Nov 15 2011 *)
%o (Magma) I:=[1,4]; [n le 2 select I[n] else 6*Self(n-1)-2*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Nov 15 2011
%Y Cf. A154244, A180028.
%K easy,nonn
%O 0,2
%A _Johannes W. Meijer_, Aug 09 2010
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