A286046 {011->1}-transform of the Thue-Morse word A010060.

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

Offset

1

Comments

From Michel Dekking, Apr 12 2022: (Start)

This sequence is a morphic sequence, i.e., the letter-to-letter image of the fixed point of a morphism mu. Here is a proof.

The Thue-Morse sequence x:=A010060 is a concatenation of the four return words A=011010, B=011001, C=01101001, and D=0110, of the word 011 in x. (These are the words occurring in x starting with 011, and having no other occurrences of 011 in them.) By applying the Thue-Morse morphism 0->01,1->10 to the return words one induces the derived morphism

tau: A->AB, B->CD, C->ABD, D->C.

This 4-letter morphism can be reduced to a 3-letter morphism by using that the letters A and B only occur in the pair E:=AB. This gives the morphism

nu: E->ECD, C->ED, D->C.

It appears that nu is nothing else but the ternary Morse morphism from A005679 on the alphabet {E,C,D}.

It is clear that to obtain (a(n)) from x one has to apply the morphism to x written as ABCDABD... given by

A->1010, B->1001, C->101001, D->10,

or to x written as ECDED... by applying the morphism delta given by

E->10101001, C->101001, D->10.

In the paper "Morphic words, Beatty sequences and integer images of the Fibonacci language" delta is called a decoration map. It is well-known that decorated fixed points of morphisms are morphic sequences, and the 'natural' algorithm to achieve this yields a morphism on an alphabet of 8+6+2 = 16 symbols. In this particular case one can reduce the number of symbols to 8, say {a,b,c,d,e,f,g,h}, and obtain the morphism mu

mu: a->ab, b->cd, c->ef, d->gh, e->abc, f->dgh, g->e, h->f.

Let y be the fixed point of mu starting with the letter a. Then (a(n)) = lambda(y), where the letter-to-letter map lambda is defined by

lambda: a->1, b->0, c->1, d->0, e->1, f->0, g->0, h->1.

(End)

Links

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

M. Dekking, Morphic words, Beatty sequences and integer images of the Fibonacci language, Theoretical Computer Science 809, 407-417 (2020).

Example

As a word, A010060 = 0110100110010110100101100..., and replacing each 011 by 1 gives 10101001101001101010100110101...

Mathematica

s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 9] (* Thue-Morse, A010060 *)
w = StringJoin[Map[ToString, s]]
w1 = StringReplace[w, {"011" -> "1"}]
st = ToCharacterCode[w1] - 48 (* A286046 *)
Flatten[Position[st, 0]]  (* A286047 *)
Flatten[Position[st, 1]]  (* A286048 *)

Crossrefs

Cf. A010060, A286047, A286048.

Sequence in context: A100060 A328101 A147850 * A189215 A285128 A080545

Adjacent sequences: A286043 A286044 A286045 * A286047 A286048 A286049

Keyword

nonn,easy

Author

Clark Kimberling, May 07 2017

Status

approved