A180037 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1+x)/(1-5*x-2*x^2).

1, 6, 32, 172, 924, 4964, 26668, 143268, 769676, 4134916, 22213932, 119339492, 641125324, 3444305604, 18503778668, 99407504548, 534045080076, 2869040409476, 15413292207532, 82804541856612, 444849293698124

Offset: 0

Author

Johannes W. Meijer, Aug 09 2010

Keywords

• easy
• nonn

Comments

The a(n) represent the number of n-move routes of a fairy chess piece starting in the corner and side squares (m = 1, 3, 7, 9; 2, 4, 6, 8) 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 A180032.

The sequence above corresponds to 28 red queen vectors, i.e. A[5] vector, with decimal values between 3 and 384. The central squares lead for these vectors to A180038.

For n>=1, a(n) equals the numbers of words of length n-1 on alphabet {0,1,2,3,4,5} containing no subwords 00, 11, 22 and 33. - Milan Janjic, Jan 31 2015, Oct 05 2016

a(n) equals the number of sequences over {0,1,2,3,4,5} of length n where no two consecutive terms differ by 4. - David Nacin, May 31 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, J. Int. Seq. 18 (2015) # 15.4.7.
• Index entries for linear recurrences with constant coefficients, signature (5, 2).

Programs

Magma

I:=[1,6]; [n le 2 select I[n] else 5*Self(n-1)+2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 15 2011

Maple

with(LinearAlgebra): nmax:=21; m:=1; 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},{1,6},50] (* Vincenzo Librandi, Nov 15 2011 *)

Formula

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

a(n) = 5*a(n-1) + 2*a(n-2) with a(0) = 1 and a(1) = 6.

a(n) = ((7-A)*A^(-n-1)+(7-B)*B^(-n-1))/33 with A = (-5+sqrt(33))/4 and B = (-5-sqrt(33))/4.