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%I #24 Sep 08 2022 08:45:54
%S 1,9,57,369,2385,15417,99657,644193,4164129,26917353,173996505,
%T 1124731089,7270376049,46996449561,303789825513,1963728301761,
%U 12693739287105,82053620627913,530402941628793,3428578511656497
%N Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1 + 3*x)/(1 - 6*x - 3*x^2).
%C The a(n) represent the number of n-move routes of a fairy chess piece starting in the center 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.
%C On a 3 X 3 chessboard there are 2^9 = 512 ways to explode with fury on the center square (off the center square the piece behaves like a normal queen). The red queen is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program and A180140. For the center square the 512 red queens lead to 17 red queen sequences, see the overview of red queen sequences and the crossreferences.
%C The sequence above corresponds to just one red queen vector, i.e., A[5] = [111 111 111] vector. The other squares lead for this vector to A090018.
%C This sequence belongs to a family of sequences with g.f. (1+k*x)/(1 - 6*x - k*x^2). The members of this family that are red queen sequences are A180028 (k=3; this sequence), A180029 (k=2), A015451 (k=1), A000400 (k=0), A001653 (k=-1), A180034 (k=-2), A084120 (k=-3), A154626 (k=-4) and A000012 (k=-5). Other members of this family are A123362 (k=5), 6*A030192(k=-6).
%C Inverse binomial transform of A107903.
%D Gary Chartrand, Introductory Graph Theory, pp. 217-221, 1984.
%H Vincenzo Librandi, <a href="/A180028/b180028.txt">Table of n, a(n) for n = 0..200</a>
%H Johannes W. Meijer, <a href="/A180028/a180028.jpg">The red queen sequences</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Alice_in_Wonderland_(2010_film)">Alice in Wonderland (2010 film)</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6, 3).
%F G.f.: (1+3*x)/(1 - 6*x - 3*x^2).
%F a(n) = 6*a(n-1) + 3*a(n-2) with a(0) = 1 and a(1) = 9.
%F a(n) = ((1-A)*A^(-n-1) + (1-B)*B^(-n-1))/4 with A=(-1+2*sqrt(3)/3) and B=(-1-2*sqrt(3)/3).
%F Lim_{k->infinity} a(n+k)/a(k) = (-1)^(n-1)*A108411(n+1)/(A041017(n-1)*sqrt(12) - A041016(n-1)) for n >= 1.
%p nmax:=19; m:=5; A[1]:=[0,1,1,1,1,0,1,0,1]: A[2]:=[1,0,1,1,1,1,0,1,0]: A[3]:=[1,1,0,0,1,1,1,0,1]: A[4]:=[1,1,0,0,1,1,1,1,0]: A[5]:=[1,1,1,1,1,1,1,1,1]: A[6]:=[0,1,1,1,1,0,0,1,1]: A[7]:=[1,0,1,1,1,0,0,1,1]: A[8]:=[0,1,0,1,1,1,1,0,1]: A[9]:=[1,0,1,0,1,1,1,1,0]: A:=Matrix([A[1], A[2], A[3], A[4], A[5], A[6], A[7], A[8], A[9]]): 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[{6,3},{1,9},50] (* _Vincenzo Librandi_, Nov 15 2011 *)
%o (Magma) I:=[1,9]; [n le 2 select I[n] else 6*Self(n-1)+3*Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Nov 15 2011
%Y Cf. A180140 (berserker sequences)
%Y Cf. A180032 (Corner and side squares).
%Y Cf. Red queen sequences center square [decimal value A[5]]: A180028 [511], A180029 [255], A180031 [495], A015451 [127], A152240 [239], A000400 [63], A057088 [47], A001653 [31], A122690 [15], A180034 [23], A180036 [7], A084120 [19], A180038 [3], A154626 [17], A015449 [1], A000012 [16], A000007 [0].
%K easy,nonn
%O 0,2
%A _Johannes W. Meijer_, Aug 09 2010; edited Jun 21 2013
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