A179604 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 2*x - 9*x^2 - 2*x^3).

1, 3, 15, 59, 259, 1079, 4607, 19443, 82507, 349215, 1479879, 6267707, 26552755, 112474631, 476459471, 2018296131, 8549676763, 36216937647, 153417558423, 649886909195, 2752965719491, 11661748738583, 49399962770975

Offset: 0

Author

Johannes W. Meijer, Jul 28 2010

Keywords

• easy
• nonn

Comments

The a(n) represent the number of n-move routes of a fairy chess piece starting in a given corner square (m = 1, 3, 7 or 9) on a 3 X 3 chessboard. This fairy chess piece behaves like a king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596.

The sequence above corresponds to 4 red king vectors, i.e., A[5] vectors, with decimal [binary] values 327 [1,0,1,0,0,0,1,1,1], 333 [1,0,1,0,0,1,1,0,1], 357 [1,0,1,1,0,0,1,0,1] and 453 [1,1,1,0,0,0,1,0,1]. These vectors lead for the side squares to A015448 and for the central square to A179605.

Links

• Index entries for linear recurrences with constant coefficients, signature (2,9,2).

Crossrefs

Cf. A001076, A001077, A015448, A179605, A179596.

Programs

Maple

with(LinearAlgebra): nmax:=22; m:=1; A[1]:= [0,1,0,1,1,0,0,0,0]: A[2]:= [1,0,1,1,1,1,0,0,0]: A[3]:= [0,1,0,0,1,1,0,0,0]: A[4]:=[1,1,0,0,1,0,1,1,0]: A[5]:= [1,0,1,1,0,0,1,0,1]: A[6]:= [0,1,1,0,1,0,0,1,1]: A[7]:= [0,0,0,1,1,0,0,1,0]: A[8]:= [0,0,0,1,1,1,1,0,1]: A[9]:= [0,0,0,0,1,1,0,1,0]: A:=Matrix([A[1],A[2],A[3],A[4],A[5],A[6],A[7],A[8],A[9]]): 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[{2,9,2},{1,3,15},30] (* or *) CoefficientList[ Series[ (x+1)/(-2 x^3-9 x^2-2 x+1),{x,0,30}],x] (* Harvey P. Dale, Mar 17 2012 *)

Formula

G.f.: ( -1-x ) / ( (2*x+1)*(x^2 + 4*x - 1) ).

a(n) = 2*a(n-1) + 9*a(n-2) + 2*a(n-3) with a(0)=1, a(1)=3 and a(2)=15.

a(n) = (20*(-1/2)^(-n) + (5+7*sqrt(5))*A^(-n-1) + (5-7*sqrt(5))*B^(-n-1))/110 with A = (-2+sqrt(5)) and B:= (-2-sqrt(5)).

Limit_{k->oo} a(n+k)/a(k) = (-1)^(n+1)/(A001076(n)*sqrt(5) - A001077(n)).