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A156595
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Fixed point of the morphism 0->011, 1->010.
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6
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0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0
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OFFSET
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0,1
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COMMENTS
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This sequence draws the Sierpinski gasket, when iterating the following odd-even drawing rule: If "1" then draw a segment forward, if "0" then draw a segment forward and turn 120 degrees right if in odd position or left if in even position.
From Dimitri Hendriks, Jun 29 2010: (Start)
This sequence is the first difference of the Mephisto Waltz A064990, i.e., a(n) = A064990(n) + A064990(n+1), where '+' is addition modulo 2.
This sequence can also be generated as a Toeplitz word: First consider the periodic word 0,1,$,0,1,$,0,1,$,... and then fill the gaps $ by the bitwise negation of the sequence itself: 0,1,_1_,0,1,_0_,0,1,_0_,.... See the Allouche/Bacher reference for a precise definition of Toeplitz sequences. (End)
From Joerg Arndt, Jan 21 2013: (Start)
Identical to the morphism 0-> 011010010, 1->011010011 given on p.100 of the Fxtbook (see link), because 0 -> 011 -> 011010010 and 1 -> 010 -> 011010011.
This sequence gives the turns (by 120 degrees) of the R9-dragon curve (displayed on p.101) which can be rendered as follows:
[Init] Set n=0 and direction=0.
[Draw] Draw a unit line (in the current direction). Turn left/right if a(n) is zero/nonzero respectively.
[Next] Set n=n+1 and goto (draw).
(End)
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REFERENCES
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M. Lothaire, Combinatorics on words.
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LINKS
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Table of n, a(n) for n=0..104.
J.-P. Allouche and R. Bacher, Toeplitz Sequences, Paperfolding, Towers of Hanoi, and Progression-Free Sequences of Integers, L'Enseignement Mathématique, volume 38, pages 315-327, 1992.
Joerg Arndt, Matters Computational (The Fxtbook) (section 1.31.5 "Dragon curves based on radix-R counting", pp. 95-101, image on p. 101).
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FORMULA
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Start with 0 and apply the morphism 0->011 and 1->010 repeatedly.
a(3k-2)=0, a(3k-1)=1, a(3k)=1-a(k) for k>=1, a(0)=0. - Clark Kimberling, Apr 28 2011
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EXAMPLE
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0 -> 0,1,1 -> 0,1,1,0,1,0,0,1,0 -> ...
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MATHEMATICA
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Nest[ Flatten[ # /. {0 -> {0, 1, 1}, 1 -> {0, 1, 0}}] &, {0}, 10]
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CROSSREFS
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Cf. A189628.
Cf. A307672 (draws curves that align with the Sierpinski gasket).
Sequence in context: A295890 A284622 A215581 * A286493 A189084 A143222
Adjacent sequences: A156592 A156593 A156594 * A156596 A156597 A156598
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KEYWORD
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easy,nice,nonn
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AUTHOR
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Alexis Monnerot-Dumaine (alexis.monnerotdumaine(AT)gmail.com), Feb 10 2009
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STATUS
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approved
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