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

[Neil Sloane]

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”.