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A180036 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1 - 2*x)/(1 - 5*x - 3*x^2).

%I A180036 #17 Sep 08 2022 08:45:54
%S A180036 1,3,18,99,549,3042,16857,93411,517626,2868363,15894693,88078554,
%T A180036 488076849,2704619907,14987330082,83050510131,460214540901,
%U A180036 2550224234898,14131764797193,78309496690659,433942777844874
%N A180036 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1 - 2*x)/(1 - 5*x - 3*x^2).
%C A180036 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 A180036 The sequence above corresponds to 56 red queen vectors, i.e., A[5] vector, with decimal values varying between 7 and 448. The corner and side squares lead for these vectors to A180035.
%H A180036 Vincenzo Librandi, <a href="/A180036/b180036.txt">Table of n, a(n) for n = 0..200</a>
%H A180036 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5, 3).
%F A180036 G.f.: (1-2*x)/(1 - 5*x - 3*x^2).
%F A180036 a(n) = 5*a(n-1) + 3*a(n-2) with a(0) = 1 and a(1) = 3.
%F A180036 a(n) = ((1+16*A)*A^(-n-1) + (1+16*B)*B^(-n-1))/37 with A = (-5+sqrt(37))/6 and B = (-5-sqrt(37))/6.
%F A180036 a(n) = Sum_{k=0..n} A202395(n,k)*2^k. - _Philippe Deléham_, Dec 21 2011
%p A180036 with(LinearAlgebra): nmax:=21; m:=5; A[5]:= [0,0,0,0,0,0,1,1,1]: 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 A180036 LinearRecurrence[{5,3},{1,3},201] (* _Vincenzo Librandi_, Nov 15 2011 *)
%o A180036 (Magma) I:=[1,3]; [n le 2 select I[n] else 5*Self(n-1)+3*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Nov 15 2011
%K A180036 easy,nonn
%O A180036 0,2
%A A180036 _Johannes W. Meijer_, Aug 09 2010
%E A180036 Second formula corrected by _Vincenzo Librandi_, Nov 15 2011