%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 oddeven 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 R9dragon 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/seals59494">Toeplitz Sequences, Paperfolding, Towers of Hanoi, and ProgressionFree Sequences of Integers</a>, L'Enseignement MathÃ©matique, volume 38, pages 315327, 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 radixR counting", pp. 95101, image on p. 101).
%F Start with 0 and apply the morphism 0>011 and 1>010 repeatedly.
%F a(3k2)=0, a(3k1)=1, a(3k)=1a(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 MonnerotDumaine (alexis.monnerotdumaine(AT)gmail.com), Feb 10 2009
