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A156595 Fixed point of the morphism 0->011, 1->010. 6

%I

%S 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,

%T 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,

%U 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

%N Fixed point of the morphism 0->011, 1->010.

%C 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.

%C From _Dimitri Hendriks_, Jun 29 2010: (Start)

%C 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.

%C 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)

%C From _Joerg Arndt_, Jan 21 2013: (Start)

%C 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.

%C This sequence gives the turns (by 120 degrees) of the R9-dragon curve (displayed on p.101) which can be rendered as follows:

%C [Init] Set n=0 and direction=0.

%C [Draw] Draw a unit line (in the current direction). Turn left/right if a(n) is zero/nonzero respectively.

%C [Next] Set n=n+1 and goto (draw).

%C (End)

%D M. Lothaire, Combinatorics on words.

%H J.-P. Allouche and R. Bacher, <a href="http://dx.doi.org/10.5169/seals-59494">Toeplitz Sequences, Paperfolding, Towers of Hanoi, and Progression-Free Sequences of Integers</a>, L'Enseignement Mathématique, volume 38, pages 315-327, 1992.

%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a> (section 1.31.5 "Dragon curves based on radix-R counting", pp. 95-101, image on p. 101).

%F Start with 0 and apply the morphism 0->011 and 1->010 repeatedly.

%F a(3k-2)=0, a(3k-1)=1, a(3k)=1-a(k) for k>=1, a(0)=0. - _Clark Kimberling_, Apr 28 2011

%e 0 -> 0,1,1 -> 0,1,1,0,1,0,0,1,0 -> ...

%t Nest[ Flatten[ # /. {0 -> {0, 1, 1}, 1 -> {0, 1, 0}}] &, {0}, 10]

%Y Cf. A189628.

%Y Cf. A307672 (draws curves that align with the Sierpinski gasket).

%K easy,nice,nonn

%O 0,1

%A Alexis Monnerot-Dumaine (alexis.monnerotdumaine(AT)gmail.com), Feb 10 2009

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Last modified April 23 07:37 EDT 2021. Contains 343201 sequences. (Running on oeis4.)