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A354896
A fixed point of the two-block substitution 00->001, 01->010, 10->101, 11->110
3
0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1
OFFSET
1
COMMENTS
Previous name was: Fixed point of the two-block Thue-Morse substitution 00->001, 01->010, 10->101, 11->110.
The two-block substitution kappa: 00->001, 01->010, 10->101, 11->110 has four fixed points, (a(n)) is the fixed point starting with 00.
Although iterates of kappa are undefined, we can generate the fixed point (a(n)) by iteration of a map kappa' defined by kappa'(w) = kappa(w) if w has even length, and kappa'(v) = kappa(w) if v = w0 or v = w1 has odd length.
It is an unsolved problem to show that the density of 1's is equal to 1/2.
Some more conjectures:
(I) If a word w occurs in (a(n)), then its mirror image w~ defined via 0~=1, 1~=0 also occurs in (a(n)).
(II) (a(n)) is uniformly recurrent, i.e., each word that occurs in (a(n)) occurs infinitely often, with bounded gaps.
LINKS
Michel Dekking and Mike Keane, Two-block substitutions and morphic words, arXiv:2202.13548 [math.CO], 2022.
Michel Dekking, The Thue-Morse sequence in base 3/2, arXiv:2301.13563 [math.CO], 2023.
EXAMPLE
The first seven iterates of the mapping kappa' starting with 0011:
0011
001110
001110101
001110101101
001110101101110010
001110101101110010110001101
001110101101110010110001101110001010101
PROG
(PARI) a(n) = my(ret=0, r); while(n>2, [n, r]=divrem(n, 3); n=2*n+r; ret+=!r); ret%2; \\ Kevin Ryde, Jun 11 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Dekking, Jun 11 2022
EXTENSIONS
Name changed by Michel Dekking, Feb 23 2023
STATUS
approved