A180036 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1 - 2*x)/(1 - 5*x - 3*x^2).

1, 3, 18, 99, 549, 3042, 16857, 93411, 517626, 2868363, 15894693, 88078554, 488076849, 2704619907, 14987330082, 83050510131, 460214540901, 2550224234898, 14131764797193, 78309496690659, 433942777844874

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 56 red queen vectors, i.e., A[5] vector, with decimal values varying between 7 and 448. The corner and side squares lead for these vectors to A180035.

Links

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

Index entries for linear recurrences with constant coefficients, signature (5, 3).

Formula

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

a(n) = 5*a(n-1) + 3*a(n-2) with a(0) = 1 and a(1) = 3.

a(n) = ((1+16*A)*A^(-n-1) + (1+16*B)*B^(-n-1))/37 with A = (-5+sqrt(37))/6 and B = (-5-sqrt(37))/6.

a(n) = Sum_{k=0..n} A202395(n,k)*2^k. - Philippe Deléham, Dec 21 2011

Maple

with(LinearAlgebra): nmax:=21; m:=5; A[5]:= [0, 0, 0, 0, 0, 0, 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[{5, 3}, {1, 3}, 201] (* Vincenzo Librandi, Nov 15 2011 *)

Prog

(Magma) I:=[1, 3]; [n le 2 select I[n] else 5*Self(n-1)+3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 15 2011

Crossrefs

Sequence in context: A081151 A132848 A321032 * A038158 A327828 A009021

Adjacent sequences: A180033 A180034 A180035 * A180037 A180038 A180039

Keyword

easy,nonn

Author

Johannes W. Meijer, Aug 09 2010

Extensions

Second formula corrected by Vincenzo Librandi, Nov 15 2011

Status

approved