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A180029 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1 + 2*x)/(1 - 6*x - 2*x^2).

%I A180029 #11 Sep 08 2022 08:45:54
%S A180029 1,8,50,316,1996,12608,79640,503056,3177616,20071808,126786080,
%T A180029 800860096,5058732736,31954116608,201842165120,1274961223936,
%U A180029 8053451673856,50870632491008,321330698293760,2029725454744576
%N A180029 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 A180029 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 A180029 The sequence above corresponds to 8 red queen vectors, i.e., A[5] vector, with decimal values 255, 383, 447, 479, 503, 507, 509 and 510. The other squares lead for these vectors to A135030.
%H A180029 Vincenzo Librandi, <a href="/A180029/b180029.txt">Table of n, a(n) for n = 0..200</a>
%H A180029 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6, 2).
%F A180029 G.f.: (1+2*x)/(1 - 6*x - 2*x^2).
%F A180029 a(n) = 6*a(n-1) + 2*a(n-2) with a(0) = 1 and a(1) = 8.
%F A180029 a(n) = ((5-4*A)*A^(-n-1) + (5-4*B)*B^(-n-1))/22 with A = (-3+sqrt(11))/2 and B = (-3-sqrt(11))/2.
%F A180029 Lim_{k->infinity} a(n+k)/a(k) = (-1)^(n-1)*A016116(n+1)/(A041015(n-1)*sqrt(11) - A041014(n-1)) for n >= 1.
%p A180029 with(LinearAlgebra): nmax:=19; m:=5; A[5]:= [0,1,1,1,1,1,1,1,1]: 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 A180029 LinearRecurrence[{6,2},{1,8},50 ] (* _Vincenzo Librandi_, Nov 15 2011 *)
%o A180029 (Magma) I:=[1,8]; [n le 2 select I[n] else 6*Self(n-1)+2*Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Nov 15 2011
%K A180029 easy,nonn
%O A180029 0,2
%A A180029 _Johannes W. Meijer_, Aug 09 2010