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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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)

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 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).

FORMULA

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

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

STATUS

approved

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Last modified November 27 18:22 EST 2020. Contains 338683 sequences. (Running on oeis4.)