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

Neil Sloane njasloane at gmail.com
Sun May 21 05:23:27 CEST 2017

```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)

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 Sat, May 20, 2017 at 8:13 PM, M. F. Hasler <seqfan at hasler.fr> wrote:

> On Sun, May 21, 2017 at 12:55 AM, Neil Sloane wrote:
> >
> > Maximilian,  I thought that there was no proof that your algorithm works.
>
> Yes that's true,
> sorry if that was not clear in my previous post where I said:
>
>  " *If* this never results in a maximal cubefree word...",
> and  "It would be sufficient ... is never maximal."
>
> I wanted to make precise where precisely the problem is located from
> the point of view of that construction / algorithm.
> Actually the critical points are where the length of the appended chunk
> equal 1:
> here we have decreased S to a minimal length and my algorithm will
> append w=[1], not without checking that this yields a cubefree word.
> (Which always happens so far, somehow by miracle.)
>
> Maximilian
>
> PS2: Yes, when I said " I refer to the computed terms "
> I really mean this : the result of my algorithm,
> and not the sequence as it is "theoretically" defined,
> which of course IS correct, there's nothing to show!
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>

```