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arXiv · 2412.17930

Runs in Paperfolding Sequences

Jeffrey Shallit
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Abstract

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 $n$'th run, is $2$-synchronized and hence computable by a finite automaton. As a specific consequence, we obtain the recent results of Bunder, Bates, and Arnold, in much more generality, via a different approach. We also prove results about the critical exponent and subword complexity of these run-length sequences.

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Jeffrey Shallit. 2025-01-02. Runs in Paperfolding Sequences. https://doi.org/10.1017/s0004972725100440

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