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A179607 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 2*x - 4*x^2)/(1 - 2*x - 8*x^2). 2
1, 4, 12, 56, 208, 864, 3392, 13696, 54528, 218624, 873472, 3495936, 13979648, 55926784, 223690752, 894795776, 3579117568, 14316601344, 57266143232, 229065097216, 916259340288, 3665039458304, 14660153638912, 58640622944256 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The a(n) represent the number of n-move routes of a fairy chess piece starting in the central square (m = 5) 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 just one red king vector, i.e., A[5] vector, with decimal [binary] value 325 [1,0,1,0,0,0,1,0,1]. This vectors leads for the corner squares to A083424 and for the side squares to A003947.

The inverse binomial transform of A100284 (without the first leading 1).

LINKS

Table of n, a(n) for n=0..23.

Index entries for linear recurrences with constant coefficients, signature (2, 8).

FORMULA

G.f.: (1 + 2*x - 4*x^2)/(1 - 2*x - 8*x^2).

a(n) = 2*a(n-1) + 8*a(n-2), for n >= 3, with a(0) = 1, a(1) = 4 and a(2) = 12.

a(n) = 5*(4)^(n)/6 - (-2)^(n)/3 for n >= 1 and a(0) = 1.

a(n) = 4*A083424(n-1), n>0. - R. J. Mathar, Mar 08 2021

MAPLE

with(LinearAlgebra): nmax:=24; m:=5; 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, 0, 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

Join[{1}, LinearRecurrence[{2, 8}, {4, 12}, 30]] (* Harvey P. Dale, Mar 01 2012 *)

CROSSREFS

Cf. A179597 (central square).

Sequence in context: A298680 A149421 A051195 * A149422 A149423 A295496

Adjacent sequences:  A179604 A179605 A179606 * A179608 A179609 A179610

KEYWORD

easy,nonn

AUTHOR

Johannes W. Meijer, Jul 28 2010

STATUS

approved

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Last modified September 27 08:22 EDT 2021. Contains 347689 sequences. (Running on oeis4.)