|
|
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.
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 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
|
|
|
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. A307672 (draws curves that align with the Sierpinski gasket).
|
|
KEYWORD
|
easy,nice,nonn
|
|
AUTHOR
|
Alexis Monnerot-Dumaine (alexis.monnerotdumaine(AT)gmail.com), Feb 10 2009
|
|
STATUS
|
approved
|
|
|
|