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 A156595 Fixed point of the morphism 0->011, 1->010. 7
 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 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

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Last modified February 28 03:01 EST 2024. Contains 370379 sequences. (Running on oeis4.)