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A180038 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1 - 3*x)/(1 - 5*x - 2*x^2). 5
1, 2, 12, 64, 344, 1848, 9928, 53336, 286536, 1539352, 8269832, 44427864, 238678984, 1282250648, 6888611208, 37007557336, 198815009096, 1068090160152, 5738080818952, 30826584415064, 165609083713224, 889698587396248 (list; graph; refs; listen; history; text; internal format)
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 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.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (5, 2).

FORMULA

G.f.: (1-3*x)/(1 - 5*x - 2*x^2).

a(n) = 5*a(n-1) + 2*a(n-2) with a(0) = 1 and a(1) = 2.

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.

MAPLE

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);

MATHEMATICA

LinearRecurrence[{5, 2}, {1, 2}, 50] (* Vincenzo Librandi, Nov 15 2011 *)

PROG

(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

CROSSREFS

Sequence in context: A283488 A006646 A087635 * A052896 A215128 A190155

Adjacent sequences:  A180035 A180036 A180037 * A180039 A180040 A180041

KEYWORD

easy,nonn

AUTHOR

Johannes W. Meijer, Aug 09 2010

STATUS

approved

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Last modified September 18 20:19 EDT 2020. Contains 337173 sequences. (Running on oeis4.)