# [seqfan] Re: The lex. earliest binary cube-free sequence

David Wilson davidwwilson at comcast.net
Tue May 2 03:52:32 CEST 2017

```Indeed, there is a mistake.
I will have to recompute.
Perhaps someone else will do so as well to keep me honest.

> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Robert
> Gerbicz
> Sent: Monday, May 01, 2017 3:37 PM
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Re: The lex. earliest binary cube-free sequence
>
> There were 10001 terms, and last one surely wrong as in the b-file
>  a(9998)=a(9999)=a(10000)=0.
>
> Added a quick Pari-Gp code to generate the sequence with backtracking.
>
> 2017-05-01 6:13 GMT+02:00 Neil Sloane <njasloane at gmail.com>:
>
> > Consider the set S of all (0,1}-sequences that do not contain
> > any cubes (no substring XXX). S is non-empty since it contains
> > Thue-Morse A010060 and also A285196. S is totally ordered
> > by lexicographic ordering. So by the Axiom of Choice there is a minimal
> > element.
> >
> > David Wilson's A282317 is defined to be this minimal element.
> >
> > But there is no proof, as far as I know, that the terms
> > that he has computed are correct (although they probably are correct.)
> > His sequence begins
> >
> > 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0,
> > 1, ..
> >
> > and he gives a b-file with 10000 terms.  One way that one might prove that
> > his terms are correct would be to guess some kind of recurrence
> > that matches his terms, and then to use this
> > characterization to prove that the infinite extension is
> > indeed cube-free and minimal.
> >
> > This is the kind of question that we all deal with every day: given
> > a sequence, find a rule that generates it. Here is a case when it would be
> > really nice to find a rule!
> >
> > Of course it could be that there is no rule, other than the definition.
> > But that is unlikely, given that  “God does not play dice with the
> > universe”.
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/

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