The paperfolding sequences form an uncountable class of infinite sequences over the alphabet $\{ -1, 1 \}$ that describe the sequence of folds arising from iterated folding of a piece of paper, followed by unfolding. In this note, we observe that the sequence of run lengths in such a sequence, as well as the starting and ending positions of the nth run, is $2$-synchronised and hence computable by a finite automaton. As a specific consequence, we obtain the recent results of Bunder, Bates and Arnold [‘The summed paperfolding sequence’, Bull. Aust. Math. Soc. 110 (2024), 189–198] in much more generality, via a different approach. We also prove results about the critical exponent and subword complexity of these run-length sequences.

Following up on a paper of Balamohan et al. [‘On the behavior of a variant of Hofstadter’s $q$-sequence’, J. Integer Seq. 10 (2007)], we analyze a variant of Hofstadter’s $Q$-sequence and show that its frequency sequence is 2-automatic. An automaton computing the sequence is explicitly given.

Two linear numeration systems, withcharacteristic polynomial equal to theminimal polynomial of two Pisot numbers β and γ respectively,such thatβ and γ are multiplicatively dependent, are considered. It is shown that the conversion between one system and the other oneis computable by a finite automaton. We also define a sequence of integers which is equal to the number of periodicpoints of a sofic dynamical system associated with some Parry number.