A354896 A fixed point of the two-block substitution 00->001, 01->010, 10->101, 11->110

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, Table of n, a(n) for n = 1..10000

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

Cf. A000002, A289001, A357448.

Sequence in context: A373141 A323158 A011657 * A359606 A072126 A287125

Adjacent sequences: A354893 A354894 A354895 * A354897 A354898 A354899

Keyword

nonn

Author

Michel Dekking, Jun 11 2022

Extensions

Name changed by Michel Dekking, Feb 23 2023

Status

approved