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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). 3
1, 5, 23, 109, 511, 2405, 11303, 53149, 249871, 1174805, 5523383, 25968589, 122092831, 574027205, 2698824263, 12688690429, 59656665391, 280479519605, 1318691881943, 6199911802669, 29149270463551, 137047105812005 (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 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

Table of n, a(n) for n=0..21.

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.

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);

CROSSREFS

Cf. A126473 (side squares).

Sequence in context: A238112 A109877 A336704 * A192810 A278677 A017974

Adjacent sequences:  A179595 A179596 A179597 * A179599 A179600 A179601

KEYWORD

easy,nonn

AUTHOR

Johannes W. Meijer, Jul 28 2010

STATUS

approved

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Last modified December 7 15:24 EST 2021. Contains 349581 sequences. (Running on oeis4.)