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

[M. F. Hasler]

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.