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 Z[[x]]. Therefore, the coefficient sequences of the formal power series expansions are 2-automatic. Write Q(n) (x) = Sigma(i >= 0) a(n),X-i(i) Then (Q(n)(x))(n >= 0) defines a two-dimensional coefficient sequence (a(n,i))(n,i >= 0). We prove that the coefficient sequences (a(n,i )mod 4)(n >= 0) introduced by both (Q(n)(x))(n >= 0) and (Pn(x))(n >= 0) are 2-automatic for all i >= 0. Moreover, the pictures of these two dimensional coefficient sequences modulo 4 present a kind of self-similar phenomenon. (C) 2020 Elsevier Inc. All rights reserved.