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 $P_n(x)\mathrm{/}{Q}_n(x)$ 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 $(a_{n,i}\mod 4){n\ge 0}$introducedbyboth$(Q{n}(x)){n\ge 0}$and$(P{n}(x))_{n\ge 0}$are$2$-automaticforall$i\ge 0$. Moreover, the pictures of these two dimensional coefficient sequences modulo $4$ present a kind of self-similar phenomenon.