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.

中文翻译：

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与折纸序列和 Rudin-Shapiro 序列相关的 Stieltjes 连分数

我们研究了由折纸序列和 Rudin-Shapiro 序列给出的两个 Stieltjes 连分数。通过明确描述收敛$P_n(x)/Q_n(x)$模$4$的某些子序列，我们给出了这两个连分数的形式幂级数展开（模$4$）并证明它们模$4$是全等的$\mathbb{Z}[[x]]$ 中的代数级数。因此，形式幂级数展开的系数序列是 $2$-自动的。写 $Q_{n}(x)=\sum_{i\ge 0}a_{n,i}x^{i}$。然后$(Q_{n}(x))_{n\ge 0}$定义了一个二维系数序列$(a_{n,i})_{n,i\ge 0}$。我们证明了由 $(Q_{n}(x))_{n\ge 0}$ 和 $( P_{n}(x))_{n\ge 0}$ 是 $2$-对于所有 $i\ge 0$ 都是自动的。而且，