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

[David Wilson]

If you concatenate the A282317 b-file elements with the Thue-Morse sequence, the resulting sequence

a = (A282317(0..10000), A010060)

has a very interesting property, specifically

- The last time the contiguous elements 0,0,1,0,0,1 occur are at a(9995..10000). 
You can verify this by proving that 0,0,1,0,0,1 are never contiguous elements of A010060.

- The first time the contiguous elements 1,0,1,1,0,1 occur are at a(10000..10005).
You can show this by showing that 1,0,1,1,0,1 are never contiguous elements of A282317(0..10000).

If a were cubefree, this would confirm that A282317(0..10000) are indeed the correct initial elements of the lexically first cubefree sequence over alphabet {0,1}, as advertised.

In attempting to prove this, the above facts would be useful in eliminating many possible cubes.
For example, suppose a had cubic subsequence XYZ with X = Y = Z.
If either X or Y included a(9995..10000), it would include the subsequence 0,0,1,0,0,1, hence so would Z.
However, Z is entirely within A010060, so it can't include subsequence 0,0,1,0,0,1.
Hence there is no cube XYZ where X or Y includes a(9995..10000).

Likewise, if Y or Z included a(10000..10005), it would include subsequence 1,0,1,1,0,1, hence so would X.
But X is entirely within A282317(0..10000), so it can't include subsequence 1,0,1,1,0,1.
Therefore there is no cube XYZ where Y or Z includes a(10000..10005).