The Thue-Morse sequence is constructed by replacing each instance of $0$ with the string $01$, and replacing $1$ with $10$. Equivalently, we can construct the sequence by successively appending the Boolean complement of the sequence obtained thus far. As an example, from $0110$ we get $01101001$, then $0110100110010110$. The Thue-Morse sequence is the sequence which these processes converge to.
There are $8$ possible $3$-bit substrings, $000,001,010,011,100,101,110,111$. Two of these are not possible, which are they and why?
How many substrings of length $4$, $5$, or in general, $n$, are there? With proof! (Obviously $4$-strings or $5$-strings that have any of the not allowed $3$-strings from the first part cannot appear. I do not have a full proof of all the substrings that can appear.)