A357448 Fixed point starting with 0 of the two-block substitution 00->010, 01->010, 10->101, 11->101.

0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1

Offset

0

Comments

Although iterates of kappa: 00->010, 01->010, 10->101, 11->101 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.

Property: If a word w occurs in (a(n)), then its binary complement w~ defined by 0~=1, 1~=0 , also occurs in (a(n)). This is proved by a check for all w of length smaller or equal to 6, and then applying induction based on kappa(w~) = (kappa(w))~.

Conjectures:

(I) (a(n)) is uniformly recurrent, i.e., each word that occurs in (a(n)) occurs infinitely often, with bounded gaps.

(II) The frequencies mu[w] of the words w occurring in (a(n)) exist. Some conjectured values: mu[00] = 1/10, mu[01] = 4/10. Moreover, it appears that mu is invariant for taking binary complements.

Links

Table of n, a(n) for n=0..88.

Michel Dekking, The Thue-Morse sequence in base 3/2, arXiv:2301.13563 [math.CO], 2023. See also J. Int. Seq., Vol. 26 (2023), Article 23.2.3.

Formula

a(n) = A244040(n) mod 2.

a(3n) = a(2n), a(3n+1) = 1-a(2n), a(3n+2) = a(2n).

Example

010010 -> 010010101-> 010010101101-> ....

Crossrefs

Cf. A354896, A244040. The base 2 version: A010060.

Sequence in context: A189572 A287028 A327202 * A083651 A111748 A282244

Adjacent sequences: A357445 A357446 A357447 * A357449 A357450 A357451

Keyword

nonn,base

Author

Michel Dekking, Sep 29 2022

Status

approved