On the automaticity of sequences defined by continued fractions

Continued fraction expansions and Hankel determinants of automatic sequences are extensively studied during the last two decades. These studies found applications in number theory in evaluating irrationality exponents. The present paper is motivated by the converse problem: to study continued fractions of which the elements form an automatic sequence. We consider two such continued fractions defined by the Thue-Morse and period-doubling sequences respectively, and prove that they are congruent to algebraic series in $\mathbb{Z}[[x]]$ modulo $4$. Consequently, the sequences of the coefficients of the power series expansions of the two continued fractions modulo $4$ are $2$-automatic. Our approach is to first guess the explicit formulas of certain subsequences of $(P_n(x))$ and $(Q_n(x))$, where $P_n(x)/Q_n(x)$ is the canonical representation of the truncated continued fractions, then prove these formulas by an intricate induction involving eight subsequences while exploiting the relations between these subsequences.