 Rudin-Shapiro sequence

For a nonnegative integer $n$, let $r(n)$ count overlapping occurrences of $11$ in its binary expansion. The Rudin-Shapiro sign sequence is $b(n)=(-1{)}^{r(n)}$. Appending a binary digit gives $b(2n)=b(n)$ and $b(2n+1)=(-1{)}^{n}b(n)$.