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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).
3
1, 8, 50, 316, 1996, 12608, 79640, 503056, 3177616, 20071808, 126786080, 800860096, 5058732736, 31954116608, 201842165120, 1274961223936, 8053451673856, 50870632491008, 321330698293760, 2029725454744576
OFFSET
0,2
COMMENTS
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.
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.
FORMULA
G.f.: (1+2*x)/(1 - 6*x - 2*x^2).
a(n) = 6*a(n-1) + 2*a(n-2) with a(0) = 1 and a(1) = 8.
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.
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.
MAPLE
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);
MATHEMATICA
LinearRecurrence[{6, 2}, {1, 8}, 50 ] (* Vincenzo Librandi, Nov 15 2011 *)
PROG
(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
CROSSREFS
Sequence in context: A240050 A221478 A287812 * A357479 A133129 A103458
KEYWORD
easy,nonn
AUTHOR
Johannes W. Meijer, Aug 09 2010
STATUS
approved