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

[Neil Sloane]

I should have been clearer.  Here is my plan of attack for showing that the
proposed 10000 terms of A282317 are correct.
Let P be a factor (subword) of these 10000 terms.
Let T be Thue-Morse.
Write P = UV and T = VW, where V is the overlap (possibly empty)
Consider the infinite sequence S = UVW.
I conjecture that S cannot be written as S = XXXZ  (use fact that T is
overlap-free?)
This would be enough to show that any prefix of the 10000-term A282317 can
be extended to an infinite cube-free sequence, and thus would show that
these 10000 terms are correct.


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.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com


On Sun, May 21, 2017 at 8:13 AM, ALLOUCHE Jean-Paul <
Jean-paul.ALLOUCHE at imj-prg.fr> wrote:

> 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.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> 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/
> >
>
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