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A285966 {11->1}-transform of the Thue-Morse word A010060. 3
0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1

COMMENTS

From Michel Dekking, Mar 15 2019: (Start)

This sequence is a morphic sequence, i.e., a letter-to-letter image of a fixed point of a morphism. Coding:

      0 followed by 0 by 0,

      0 followed by 1 by 2,

      1 by 1 if 1 was in 010 in the TM word,

      1 by 3 if it was the image of 11 in the TM word.

It then follows that (a(n)) is the letter to letter image 0->0, 1->1, 2->0, 3->1 of the unique fixed point of the morphism

      0 -> 21

      1 -> 0

      2 -> 23

      3 -> 210.

Explanation of (the more difficult) first and last productions:

  010 -> 011001 by the TM morphism, which is coded to 2302. Here the 3 is put in the production 2 -> 23, the 0 is from the production 1 -> 0, and the final 2 is taken care of in either the production 0 -> 21, or the production 2 -> 23. For the last production, note that 0110 -> 01101001 by the TM morphism, which is coded to 232102. Here the prefix 23 is taken care of by the production 2 -> 23, then 11 coded by 3 gives 210, and the final 2 comes from the final 0 in 0110.

(End)

LINKS

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

EXAMPLE

As a word, A010060 = 0110100110010110100101100..., and replacing each 11 by 1 gives 01010010010101001010010100100101...

MATHEMATICA

s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 9] (* Thue-Morse, A010060 *)

w = StringJoin[Map[ToString, s]]

w1 = StringReplace[w, {"11" -> "1"}]

st = ToCharacterCode[w1] - 48 (* A285966 *)

Flatten[Position[st, 0]]  (* A285967 *)

Flatten[Position[st, 1]]  (* A285968 *)

CROSSREFS

Cf. A010060, A285967, A285968.

Sequence in context: A141474 A073424 A135993 * A215530 A241422 A189661

Adjacent sequences:  A285963 A285964 A285965 * A285967 A285968 A285969

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, May 06 2017

STATUS

approved

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Last modified July 20 22:20 EDT 2019. Contains 325189 sequences. (Running on oeis4.)