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
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).
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 *)
PROG
(Python)
from sympy import integer_log
def A156595(n): return sum(((m:=(n+1)//9**i)-2)//9+(m-3)//9+(m-5)//9+(m-8)//9+4 for i in range(integer_log(n+1, 9)[0]+1))-sum(((m:=n//9**i)-2)//9+(m-3)//9+(m-5)//9+(m-8)//9+4 for i in range(integer_log(n, 9)[0]+1)) if n else 0 # Chai Wah Wu, Feb 16 2025
CROSSREFS
KEYWORD
easy,nice,nonn
AUTHOR
Alexis Monnerot-Dumaine, Feb 10 2009
STATUS
approved