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The Rudin-Shapiro sequence is defined as follows:

Let $a_n=\sum\epsilon_i\epsilon_{i+1}$ where $\epsilon_1,\epsilon_2...$ are the digits in the binary expansion of $n$. $WS(n)$, the $n$th term of the Rudin-Shapiro sequence, is defined as$$WS(n)=(-1)^{a_n}$$

My question is: Is there a closed form of this sequence? I know that it can be defined recursively as$$\begin{cases}r_{2n}&=r_n\\r_{2n+1}&=(-1)^nr_n\end{cases}$$however there doesn't seem to be a closed form of this sequence that I'm aware of, and while there does seem to be a generalized form for it ($||P||_\infty\le\sqrt2||P||_2$), I haven't been able to find a closed form.

My motivation is that I got interested in the sequence a while ago and have been trying to find a closed form of it using its recursive definition.

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    $\begingroup$ $$\begin{cases}r_{2n}&=r_n\\r_{2n+1}&=(-1)^nr_n\end{cases}$$ is a recursion formula for $(-1)^{t_n}$ where $(t_n)$ is the Thue-Morse sequence, not for the Rudin-Shapiro sequence. $\endgroup$
    – Christophe Leuridan
    Dec 19 at 12:07
  • $\begingroup$ @ChristopheLeuridan Oh crap I must have mixed the two up let me fix that quick $\endgroup$
    – CrSb0001
    Dec 19 at 12:08
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    $\begingroup$ OEIS does not list a closed form, I presume the answer is "no" ---oeis.org/A020985 $\endgroup$
    – Carlo Beenakker
    Dec 19 at 13:51

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