Stieltjes continued fractions related to the Paperfolding sequence and Rudin-Shapiro sequence

We investigate two Stieltjes continued fractions given by the paperfolding sequence and the Rudin-Shapiro sequence. By explicitly describing certain subsequences of the convergents $P_n(x)/Q_n(x)$ modulo $4$, we give the formal power series expansions (modulo $4$) of these two continued fractions and prove that they are congruent modulo $4$ to algebraic series in $\mathbb{Z}[[x]]$. Therefore, the coefficient sequences of the formal power series expansions are $2$-automatic. Write $Q_{n}(x)=\sum_{i\ge 0}a_{n,i}x^{i}$. Then $(Q_{n}(x))_{n\ge 0}$ defines a two-dimensional coefficient sequence $(a_{n,i})_{n,i\ge 0}$. We prove that the coefficient sequences $(a_{n,i}\mod 4)_{n\ge 0}$ introduced by both $(Q_{n}(x))_{n\ge 0}$ and $(P_{n}(x))_{n\ge 0}$ are $2$-automatic for all $i\ge 0$. Moreover, the pictures of these two dimensional coefficient sequences modulo $4$ present a kind of self-similar phenomenon.

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