Canonical positions for the factors in paperfolding sequences

Abstract

It has been proved by the first author that the number of factors of length k in any paperfolding sequence is equal to 4 k once k⩾7. We prove here that for every integer k, there exists a set of integers P k of cardinality 4 k for k ⩾ 7 such that the factors of length k of any paperfolding sequence are exactly the factors beginning at the places indexed by the integers in P k : this gives a “bijective” proof of the result mentioned above. Then we give an explicit uniform linear upper bound for the recurrence function of paperfolding sequences. Finally, we study the set of all factors of all paperfolding sequences, evaluating their number for a given length, and studying the language of these words.