I'm not sure if "concatenated" or "contiguous" is the better word here, or what the best way of phrasing it is, so let me give an example.
Consider strings of $\ 0'$s and $\ 1'$s, such as $\ '1000110101'.$ What is (the length, $n,$ of) the longest string that has no $k$ contiguous repeat substrings? For example, for $k=2,\ $ we are not allowed any substrings to repeat more than once. So, $\ '110'\ $ would fail, as it contains the substring $\ '11' = \,'1' + \,'1'.\ $ And $\ '10101'\ $ would fail, as it contains the substring $\ '1010' = \,'10' + \,'10'.\ $ So the longest substrings for $k=2$ are $\,'101'$ or $\,'010'$, and so for $k=2,$ our longest string has length $n=3.$
However, for $k=3,$ things already get seemingly wild: I think $\,'11011010110110'$ doesn't fail, and we can append many more $0's$ and $1's$ to the end before it fails.
What is the largest value of $n$ for a given $k$?
Is this a known problem? If not, can we make any progress on it?
I am not asking for de Bruijn sequences, although these might be related to my question somehow.