

A156595


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


6



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

0,1


COMMENTS

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.
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 R9dragon 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)


REFERENCES

M. Lothaire, Combinatorics on words.


LINKS

Table of n, a(n) for n=0..104.
J.P. Allouche and R. Bacher, Toeplitz Sequences, Paperfolding, Towers of Hanoi, and ProgressionFree Sequences of Integers, L'Enseignement MathÃ©matique, volume 38, pages 315327, 1992.
Joerg Arndt, Matters Computational (The Fxtbook) (section 1.31.5 "Dragon curves based on radixR counting", pp. 95101, image on p. 101).


FORMULA

Start with 0 and apply the morphism 0>011 and 1>010 repeatedly.
a(3k2)=0, a(3k1)=1, a(3k)=1a(k) for k>=1, a(0)=0.  Clark Kimberling, Apr 28 2011


EXAMPLE

0 > 0,1,1 > 0,1,1,0,1,0,0,1,0 > ...


MATHEMATICA

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


CROSSREFS

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


KEYWORD

easy,nice,nonn


AUTHOR

Alexis MonnerotDumaine (alexis.monnerotdumaine(AT)gmail.com), Feb 10 2009


STATUS

approved



