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A171587
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Sequence of the diagonal variant of the Fibonacci word fractal. Sequence of the Fibonacci tile.
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2
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0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0
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OFFSET
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0,1
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COMMENTS
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Interpreted as 0=turn right and 1=turn left, this sequence builds the diagonal variant of the Fibonacci word fractal. Base for the construction of the Fibonacci tile (Tiles the plane by translation in 2 ways).
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REFERENCES
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A. Blondin-Masse, S. Brlek, A. Garon, S. Labbe, Christoffel and Fibonacci tiles, sept 2009.
A. Monnerot-Dumaine, The Fibonacci word fractal, feb 2009
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LINKS
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Table of n, a(n) for n=0..104.
A. Blondin-Masse, S. Brlek, A. Garon, S. Labbe, Christoffel and Fibonacci Tiles
A. Monnerot-Dumaine, The Fibonacci word fractal
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FORMULA
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a(n) = a(n-1).a(n-2) if n= 2 mod 3, else a(n) = a(n-1).bar{a(n-2)}, with a(0) the empty word and a(1)=0, where bar exchanges 0 and 1. Also application of the mapping 1->0, 2->1, 0->emptyword to the Dense Fibonacci word A143667.
Conjecture: A171587=(A001950 mod 2), as suggested for n=1,2,...,500 by Mathematica program below. [From Clark Kimberling, May 31 2011]
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EXAMPLE
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a(2) = a(1).a(0) = 0, a(3) = a(2).bar{a(1)} = 01, a(4) = a(3).bar{a(2)} = 011, a(5) = a(4).a(3) = 01101.
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MATHEMATICA
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(* This program supports the conjecture that A171587=(A001950 mod 2). *)
t = Nest[Flatten[# /. {1 -> {1, 0, 2, 2}, 0 -> {1, 0, 2, 2, 1}, 2 -> {1, 0, 2, 1}}] &, {1}, 5]
w = DeleteCases[t, 0] /. {1 -> 0, 2 -> 1}
u = Table[n + Floor[n*GoldenRatio], {n, 1, 500}]; v = Mod[u, 2]
Table[w[[n]] - v[[n]], {n, 1, 500}] (* supports conjecture for n=1, 2, ..., 500 *)
(* t=A143667, w=A171587, u=A001950, conjecture: v=w *)
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CROSSREFS
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Cf. A143667, A003849;
A001950 (upper Wythoff sequence),
A085002 (lower Wythoff sequence mod 2).
Sequence in context: A103226 A080886 A083924 * A082410 A094217 A174784
Adjacent sequences: A171584 A171585 A171586 * A171588 A171589 A171590
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KEYWORD
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nonn
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AUTHOR
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Alexis Monnerot-Dumaine (alexis.monnerotdumaine(AT)gmail.com), Dec 12 2009
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STATUS
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approved
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