A285952 {0->1, 1->10}-transform of the Thue-Morse word A010060.

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

Offset

1

Comments

The morphism {0->1, 1->10} has the infinite Fibonacci word A005614 as fixed point.

Since the Thue-Morse word is a concatenation of the 2-blocks 01 and 10, this word is a concatenation of 3-blocks 110 and 101. In particular the frequency of 0 is 1/3. This also follows from the fact that this sequence is a morphic sequence, i.e., the letter-to-letter image of the fixed point of a morphism. The morphism on {1,2,3} can be chosen as 1->123, 2->23, 3->1, and the letter-to-letter map as 1->1, 2->1, 3->0. - Michel Dekking, Sep 20 2020

Actually this sequence is an automatic sequence. It follows directly from the observations above that (a(n)) is the decoration by the map 0->110, 1->101 of the Thue-Morse word A010060. - Michel Dekking, Jun 18 2022

Links

Table of n, a(n) for n=1..86.

Example

As a word, A010060 = 0110100110010110100101100..., and replacing each 0 by 1 and each 1 by 10 gives 110101101110101110110101...

Mathematica

s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 7]  (* Thue-Morse, 010060 *)
w = StringJoin[Map[ToString, s]]
w1 = StringReplace[w, {"0" -> "1", "1" -> "10"}]  (* A285952, word *)
st = ToCharacterCode[w1] - 48 (* A285952, sequence *)
Flatten[Position[st, 0]]  (* A285953 *)
Flatten[Position[st, 1]]  (* A285954 *)

Crossrefs

Cf. A010060, A005614, A285950, A285953, A285954.

Sequence in context: A267269 A187972 A248396 * A371691 A103842 A286064

Adjacent sequences: A285949 A285950 A285951 * A285953 A285954 A285955

Keyword

nonn,easy

Author

Clark Kimberling, May 05 2017

Status

approved