A179600 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 2*x - 10*x^2 - 4*x^3).

1, 3, 16, 66, 304, 1332, 5968, 26472, 117952, 524496, 2334400, 10385568, 46213120, 205619520, 914912512, 4070872704, 18113348608, 80595074304, 358607125504, 1595618388480, 7099688329216, 31589989045248, 140559334936576

Offset

0,2

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 6 red king vectors, i.e., A[5] vectors, with decimal values 335, 359, 365, 455, 461 and 485. These vectors lead for the side squares to A123347 and for the central square to A179601.

Links

Table of n, a(n) for n=0..22.

Index entries for linear recurrences with constant coefficients, signature (2,10,4).

Formula

G.f.: (1+x)/(1 - 2*x - 10*x^2 - 4*x^3).

a(n) = 2*a(n-1) + 10*a(n-2) + 4*a(n-3) with a(0)=1, a(1)=3 and a(2)=16.

a(n) = (4*(-1/2)^(-n) + (1+sqrt(6))*A^(-n-1) + (1-sqrt(6))*B^(-n-1))/20 with A = (-1+sqrt(6)/2) and B = (-1-sqrt(6)/2).

Lim_{k->infinity} a(n+k)/a(k) = (-1)^(n+1)*(A016116(n+1)/(A041007(n-1)*sqrt(6) - A041006(n-1))) for n => 1.

Maple

with(LinearAlgebra): nmax:=24; 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, 1, 1, 0, 0, 0, 1, 1, 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);

Prog

(PARI) Vec((1+x)/(1 - 2*x - 10*x^2 - 4*x^3) + O(x^40)) \\ Jinyuan Wang, Mar 10 2020

Crossrefs

Sequence in context: A062960 A293579 A044046 * A278089 A248016 A000269

Adjacent sequences: A179597 A179598 A179599 * A179601 A179602 A179603

Keyword

nonn,easy

Author

Johannes W. Meijer, Jul 28 2010

Status

approved