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

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

Offset

0,1

Comments

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

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).

Gabriele Fici and Jeffrey Shallit, Properties of a Class of Toeplitz Words, arXiv:2112.12125 [cs.FL], 2021.

Kevin Ryde, Iterations of the Terdragon Curve, see index "AltTurnRpred" with AltTurnRpred(n) = a(n-1).

Index entries for sequences that are fixed points of mappings

Formula

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]
SubstitutionSystem[{0->{0, 1, 1}, 1->{0, 1, 0}}, 0, {5}][[1]] (* Harvey P. Dale, Jan 15 2022 *)

Crossrefs

Cf. A278996 (indices of 0's), A278997 (indices of 1's), A189717 (partial sums).

Cf. A189628 (morphisms guide).

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

Sequence in context: A342704 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