A179601 - OEIS

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A179601

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

1, 6, 22, 108, 460, 2088, 9208, 41136, 182704, 813600, 3618784, 16104384, 71651008, 318820992, 1418569600, 6311953152, 28084886272, 124963582464, 556023840256, 2474023050240, 11008138832896, 48980603529216, 217938687588352 (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 6 red king vectors, i.e., A[5] vectors, with decimal values 335, 359, 365, 455, 461 and 485. These vectors lead for the corner squares to A179600 and for the side squares to A123347.`
	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.: ( -1-4x ) / ( (2x+1)(2x^2 + 4x - 1) ).` `a(n) = 2a(n-1) + 10a(n-2) + 4a(n-3) with a(0)=1, a(1)=6 and a(2)=22.` `a(n) = (-2/5)(-1/2)^(-n) + ((2+3A)A^(-n-1) + (2+3B)B^(-n-1))/10 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:=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]:= [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);
	CROSSREFS	`Cf. A179597 (central square).` `Sequence in context: A078418 A100300 A027296 * A151495 A193463 A319214` `Adjacent sequences: A179598 A179599 A179600 * A179602 A179603 A179604`
	KEYWORD	`easy,nonn`
	AUTHOR	`Johannes W. Meijer, Jul 28 2010`
	STATUS	`approved`

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Last modified March 25 23:47 EDT 2023. Contains 361529 sequences. (Running on oeis4.)