The ThueMorse(n, a, b) command generates the length-n prefix of the (infinite) Thue-Morse string, with initial values a and b. The arguments after the first are optional, and default to the strings $''0''$ and $''1''$, respectively.

The Thue-Morse string is the infinite string defined to be the fixed point of the iterated semigroup morphism $0->01,1->10$ on the free semigroup generated by $\left\{0\,1\right\}$.

A recursive definition of the Thue-Morse sequence is as follows. $t_0=0$, $t_{2n}=t_n$, $t_{1+2n}=1-t_n$.

All of the StringTools package commands treat strings as (null-terminated) sequences of $8$-bit (ASCII) characters.  Thus, there is no support for multibyte character encodings, such as unicode encodings.

$\mathrm{with}\left(\mathrm{StringTools}\right)\:$

$\mathrm{seq}\left(\mathrm{ThueMorse}\left(n\right)\,n=0..5\right)$

$'''',''0'',''01'',''011'',''0110'',''01101''$

$\mathrm{ThueMorse}\left(5\,''a''\,''b''\right)$

$''abbab''$

$\mathrm{ThueMorse}\left(100\right)$

$''0110100110010110100101100110100110010110011010010110100110010110100101100110100101101001100101100110''$