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A179611 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1+2*x)/(1 - 2*x - 8*x^2 - 4*x^3). 1
1, 4, 16, 68, 280, 1168, 4848, 20160, 83776, 348224, 1447296, 6015488, 25002240, 103917568, 431915008, 1795179520, 7461349376, 31011794944, 128895102976, 535729963008, 2226667929600, 9254755975168, 38465775239168 (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 36 red king vectors, i.e., A[5] vectors, with decimal values 15, 39, 45, 75, 78, 99, 102, 105, 108, 135, 141, 165, 195, 198, 201, 204, 225, 228, 267, 270, 291, 294, 297, 300, 330, 354, 360, 387, 390, 393, 396, 417, 420, 450, 456 and 480.

LINKS

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

Index entries for linear recurrences with constant coefficients, signature (2,8,4).

FORMULA

G.f.: (1+2*x)/(1 - 2*x - 8*x^2 - 4*x^3).

a(n) = 2*a(n-1) + 8*a(n-2) + 4*a(n-3) with a(1)=1, a(2)=4 and a(3)=16.

a(n) = (8 + 3*z1 - 6*z1^2)*z1^(-n)/(z1*37) + (8 + 3*z2 - 6*z2^2)*z2^(-n)/(z2*37) + (8 + 3*z3 - 6*z3^2)*z3^(-n)/(z3*37) with z1, z2 and z3 the roots of f(x) = 1 - 2*x - 8*x^2 - 4*x^3 = 0.

alpha = arctan(3*sqrt(111));

z1 = sqrt(10)*cos(alpha/3)/6 + sqrt(30)*sin(alpha/3)/6 - 2/3 = 0.2405971520460078;

z2 = -sqrt(10)*cos(alpha/3)/3 - 2/3 = -1.585043243313016;

z3 = sqrt(10)*cos(alpha/3)/6 - sqrt(30)*sin(alpha/3)/6 - 2/3 = -0.6555539087329909.

MAPLE

with(LinearAlgebra): nmax:=22; 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]:= [0, 0, 0, 0, 0, 1, 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);

MATHEMATICA

LinearRecurrence[{2, 8, 4}, {1, 4, 16}, 30] (* Harvey P. Dale, Oct 20 2017 *)

CROSSREFS

Cf. A179597 (central square).

Cf. A052904.

Sequence in context: A283036 A307051 A158761 * A290912 A089979 A179191

Adjacent sequences:  A179608 A179609 A179610 * A179612 A179613 A179614

KEYWORD

easy,nonn

AUTHOR

Johannes W. Meijer, Jul 28 2010

STATUS

approved

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Last modified September 27 08:39 EDT 2021. Contains 347689 sequences. (Running on oeis4.)