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

[Neil Sloane]

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.
>
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