Generalized Fermat, double Fermat and Newton sequences

In this paper, we discuss the relationship among the generalized Fermat, double Fermat, and Newton sequences. In particular, we show that every double Fermat sequence is a generalized Fermat sequence, and the set of generalized Fermat sequences, as well as the set of double Fermat sequences, is closed under term-by-term multiplication. We also prove that every Newton sequence is a generalized Fermat sequence and vice versa. Finally, we show that double Fermat sequences are Newton sequences generated by certain sequences of integers. An approach of symbolic dynamical systems is used to obtain congruence identities.