A179598 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 2*x)/(1 - 3*x - 8*x^2).

1, 5, 23, 109, 511, 2405, 11303, 53149, 249871, 1174805, 5523383, 25968589, 122092831, 574027205, 2698824263, 12688690429, 59656665391, 280479519605, 1318691881943, 6199911802669, 29149270463551, 137047105812005

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 side square (m = 2, 4, 6 or 8) 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 10 red king vectors, i.e., A[5] vectors, with decimal values 239, 351, 375, 381, 431, 471, 477, 491, 494, and 501. These vectors lead for the corner squares to A015525 and for the central square to A179599.

Inverse binomial transform of A126501.

Links

• Index entries for linear recurrences with constant coefficients, signature (3, 8).

Crossrefs

Cf. A126473 (side squares).

Programs

Maple

with(LinearAlgebra): nmax:=21; m:=2; 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]:= [1,0,1,1,1,1,1,0,1]: 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);

Formula

G.f.: (1+2*x)/(1 - 3*x - 8*x^2).

a(n) = 3*a(n-1) + 8*a(n-2) with a(0) = 1 and a(1) = 5.

a(n) = ((41+5*sqrt(41))*A^(-n-1) + (41-5*sqrt(41))*B^(-n-1))/328 with A = (-3+sqrt(41))/16 and B = (-3-sqrt(41))/16.