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A171587 Sequence of the diagonal variant of the Fibonacci word fractal. Sequence of the Fibonacci tile. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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).

LINKS

Table of n, a(n) for n=0..104.

A. Blondin-Masse, S. Brlek, A. Garon, S. Labbe, Christoffel and Fibonacci Tiles , Sept 2009.

A. Blondin-Masse, S. Brlek, A. Garon, S. Labbe, Christoffel and Fibonacci tiles, Sept 2009.

A. Blondin-Masse, S. Brlek, A. Garon, S. Labbe, Christoffel and Fibonacci tiles presentation, Sept 2009.

A. Monnerot-Dumaine, The Fibonacci word fractal, Feb 2009.

FORMULA

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. - Clark Kimberling, May 31 2011

EXAMPLE

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.

MATHEMATICA

(* 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 *)

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Alexis Monnerot-Dumaine (alexis.monnerotdumaine(AT)gmail.com), Dec 12 2009

STATUS

approved

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Last modified March 22 22:05 EDT 2017. Contains 283901 sequences.