# [seqfan] Re: More about lex earliest cubefree 0,1 sequence

ALLOUCHE Jean-Paul Jean-paul.ALLOUCHE at imj-prg.fr
Sun May 21 14:13:05 CEST 2017

```Dear all

There is indeed a TM sequence with indexes in Z extending
the  N-indexed one. Furthermore if I am not mistaken it
has exactly the same factors (finIte blocks of consecutive
letters) as the classical TM sequence. Hence it is overlap-free
as well. So are all suffixes of the  Z-indexed TM. (I am in a train station right now and do not  have  a reference at hand.)
best
jean-paul
_______________________________________
De : SeqFan [seqfan-bounces at list.seqfan.eu] de la part de Neil Sloane [njasloane at gmail.com]
Envoyé : dimanche 21 mai 2017 13:34
À : Sequence Fanatics Discussion list
Objet : [seqfan] Re: More about lex earliest cubefree 0,1 sequence

My argument that 0010T is cubefree also shows that 0T and 1T are cubefree
(T = Thue-Morse). But how far can that argument be extended?

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Email: njasloane at gmail.com

On Sun, May 21, 2017 at 5:05 AM, M. F. Hasler <seqfan at hasler.fr> wrote:

> On Sun, May 21, 2017 at 4:23 AM, Neil Sloane wrote:
> > What I am really hoping for is a theorem that says something like this:
> > Given a finite 0,1 cubefree string S satisfying such-and-such properties,
> > then there is an infinite cubefee string that begins with S
> > (and perhaps that S can be obtained by attaching the Thue-Morse sequence
> to
> > the end of S, maybe with a bit of trimming of some overlap)
>
> Since you are not specific about the properties,
> a positive answer to this question would be given
> if we can show that either a 0 or 1 can be prefixed to the Thue-Morse
> sequence and yield again a cubefree sequence.
> Experimentally it appears that both choices would work, but I don't
> have a proof.
>
> At a first glance, it appears that it could even be possible to extend
> the Thue-Morse sequence infinitely to the left to a cube-free
> sequence.
>
> - M.
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>

--
Seqfan Mailing list - http://list.seqfan.eu/

```