repetition sequence

[Dion Gijswijt]

Hello Seqfans,

Recently I stumbled over a funny sequence that starts as follows:

1,1,2,1,1,2,2,2,3,
1,1,2,1,1,2,2,2,3,2,
1,1,2,1,1,2,2,2,3,
1,1,2,1,1,2,2,2,3,2,2,2,3,2,2,2,3,3,2

1,1,2,1,1,2,2,2,3,
1,1,2,1,1,2,2,2,3,2,
1,1,2,1,1,2,2,2,3,
1,1,2,1,1,2,2,2,3,2,2,2,3,2,2,2,3,3,2,2,2,3,2,

1,1,2,1,1,2,2,2,3,
1,1,2,1,1,2,2,2,3,2,
1,1,2,1,1,2,2,2,3,
1,1,2,1,1,2,2,2,3,2,2,2,3,2,2,2,3,3,2,

1,1,2,1,1,2,2,2,3,
1,1,2,1,1,2,2,2,3,2,
1,1,2,1,1,2,2,2,3,
1,1,2,1,1,2,2,2,3,2,2,2,3,2,2,2,3,3,2,2,2,3,2,2,2,3,2,2,2,3,3,2,2,2,3,2,2,2,3,2,2,2,3,3,3,3,4,

1,1,...

This sequence a(1),a(2),...) is defined as follows:
a(1)=1
a(n+1) is defined as the largest integer k such that the word 
a(0)a(1)...a(n) is of the form xy^k for words x and y (y of positive 
length). Here xy^k means concatenating the words x and k copies of y.
So each term counts the max. number of repeating blocks at the end of 
the sequence so far.

Note that the first 4 occurs at a(220).
An obvious question is: which numbers occur in this sequence? Is the 
sequence unbounded?

This question has puzzled me for some time now. I even went as far as 
calculation the first 300,000 numbers in the sequence. I didn't find any 
number >4, not even two adjacent 4's. Still I have the feeling that the 
sequence is unbounded, but that the sequence b(n):=first occurence of n 
is a VERY fast growing sequence.

If anyone has some idea about this, please let me know!


Best regards,

Dion Gijswijt.