On synchronized sequences and their separators

We introduce the notion of a k -synchronized sequence, where k is an integer larger than 1. Roughly speaking, a sequence of natural numbers is said to be k -synchronized if its graph is represented, in base k , by a right synchronized rational relation. This is an intermediate notion between k -automatic and k -regular sequences. Indeed, we show that the class of k -automatic sequences is equal to the class of bounded k -synchronized sequences and that the class of k -synchronized sequences is strictly contained in that of k -regular sequences. Moreover, we show that equality of factors in a k -synchronized sequence is represented, in base k , by a right synchronized rational relation. This result allows us to prove that the separator sequence of a k -synchronized sequence is a k -synchronized sequence, too. This generalizes a previous result of Garel, concerning k -regularity of the separator sequences of sequences generated by iterating a uniform circular morphism.