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A180031 Number of n-move paths on a 3 X 3 chessboard of a queen starting or ending in the central square.

%I A180031 #15 Sep 08 2022 08:45:54
%S A180031 1,8,48,304,1904,11952,74992,470576,2952816,18528688,116265968,
%T A180031 729559344,4577924464,28726097072,180253881072,1131078181936,
%U A180031 7097421958256,44535735246768,279458051899888,1753576141473584
%N A180031 Number of n-move paths on a 3 X 3 chessboard of a queen starting or ending in the central square.
%C A180031 The a(n) represent the number of n-move paths of a chess queen starting or ending in the central square (m = 5) on a 3 X 3 chessboard. The other squares lead to A180030.
%C A180031 To determine the a(n) we can either sum the components of the column vector A^n[k,m], with A the adjacency matrix of the queen's graph, or we can sum the components of the row vector A^n[m,k], see the Maple program.
%C A180031 Closely related with this sequence are the red queen sequences, see A180028 and A180032.
%C A180031 This sequence belongs to a family of sequences with g.f. (1+k*x)/(1 - 5*x - (k+5)*x^2). The members of this family that are red queen sequences are A180031 (k=3; this sequence), A152240 (k=2), A000400 (k=1), A057088 (k=0), A122690 (k=-1), A180036 (k=-2), A180038 (k=-3), A015449 (k=-4) and A000007 (k=-5). Other members of this family are A030221 (k= -6), 3*A109114 (k=-8), 4*A020989 (k=-9), 6*A166060 (k=-11).
%H A180031 Vincenzo Librandi, <a href="/A180031/b180031.txt">Table of n, a(n) for n = 0..200</a>
%H A180031 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5, 8).
%F A180031 G.f.: (1+3*x)/(1 - 5*x - 8*x^2).
%F A180031 a(n) = 5*a(n-1) + 8*a(n-2) with a(0) = 1 and a(1) = 8.
%F A180031 a(n) = ((A+11)*A^(-n-1) + (B+11)*B^(-n-1))/57 with A = (-5+sqrt(57))/16 and B = (-5-sqrt(57))/16.
%p A180031 with(LinearAlgebra): nmax:=19; m:=5; A[5]:= [1,1,1,1,0,1,1,1,1]: 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 A180031 LinearRecurrence[{5,8},{1,8},50] (* _Vincenzo Librandi_, Nov 15 2011 *)
%o A180031 (Magma) I:=[1,8]; [n le 2 select I[n] else 5*Self(n-1)+8*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Nov 15 2011
%K A180031 nonn,easy
%O A180031 0,2
%A A180031 _Johannes W. Meijer_, Aug 09 2010