 Thue–Morse sequence

Here $s_2(n)$ is the sum of the binary digits of the nonnegative integer $n$. The sign version is $f(n)=(-1{)}^{s_2(n)}$. Concatenating binary digits gives $f(a{2}^j+r)=f(a)f(r)$ for $0\le r<2^j$. It supplies cancellation controlled by binary blocks rather than multiplicative factorization.