A180035 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1+x)/(1-5*x-3*x^2).

1, 6, 33, 183, 1014, 5619, 31137, 172542, 956121, 5298231, 29359518, 162692283, 901539969, 4995776694, 27683503377, 153404846967, 850074744966, 4710588265731, 26103165563553, 144647592614958, 801547459765449

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 56 red queen vectors, i.e. A[5] vector, with decimal values between 7 and 448. The central squares lead for these vectors to A180036.

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, example 17.
• Index entries for linear recurrences with constant coefficients, signature (5, 3).

Programs

Magma

I:=[1,6]; [n le 2 select I[n] else 5*Self(n-1)+3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 15 2011

Maple

with(LinearAlgebra): nmax:=20; m:=1; 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},{1,6},50] (* Vincenzo Librandi, Nov 15 2011 *)

Formula

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

a(n) = 5*a(n-1) + 3*a(n-2) with a(0) = 1 and a(1) = 6.

a(n) = ((7+A)*A^(-n-1)+(7+B)*B^(-n-1))/37 with A = (-5+sqrt(37))/6 and B = (-5-sqrt(37))/6.

a(n) = Sum_{k, 0<=k<=n} A202396(n,k)*2^k. - Philippe Deléham, Dec 21 2011

a(n) = A015536(n+1)+A015536(n). - R. J. Mathar, Aug 04 2019