

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
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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 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.
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)
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



FORMULA

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


CROSSREFS

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


KEYWORD

easy,nice,nonn


AUTHOR

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


STATUS

approved



