|
%I #11 Sep 08 2022 08:45:54
%S 1,2,12,64,344,1848,9928,53336,286536,1539352,8269832,44427864,
%T 238678984,1282250648,6888611208,37007557336,198815009096,
%U 1068090160152,5738080818952,30826584415064,165609083713224,889698587396248
%N Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1 - 3*x)/(1 - 5*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 28 red queen vectors, i.e., A[5] vector, with decimal values varying between 3 and 384. The corner and side squares lead for these vectors to A180037.
%H Vincenzo Librandi, <a href="/A180038/b180038.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 (5, 2).
%F G.f.: (1-3*x)/(1 - 5*x - 2*x^2).
%F a(n) = 5*a(n-1) + 2*a(n-2) with a(0) = 1 and a(1) = 2.
%F a(n) = ((19*A-1)*A^(-n-1) + (19*B-1)*B^(-n-1))/33 with A = (-5+sqrt(33))/4 and B = (-5-sqrt(33))/4.
%p with(LinearAlgebra): nmax:=21; m:=5; A[5]:= [0,0,0,0,0,0,0,1,1]: 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[{5,2},{1,2},50] (* _Vincenzo Librandi_, Nov 15 2011 *)
%o (Magma) I:=[1,2]; [n le 2 select I[n] else 5*Self(n-1)+2*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Nov 15 2011
%K easy,nonn
%O 0,2
%A _Johannes W. Meijer_, Aug 09 2010
|
