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

Robert Gerbicz robert.gerbicz at gmail.com
Mon May 1 21:36:50 CEST 2017

```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/
>

```

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