Raghavendra N. Bhat, Cristian Cobeli, Alexandru Zaharescu: On quasi-periodicity in Proth-Gilbreath triangles, 3-21

Abstract:

Let $\operatorname{PG}$ be the Proth-Gilbreath operator that transforms a sequence of integers into the sequence of the absolute values of the differences between all pairs of neighbor terms. Consider the infinite tables obtained by successive iterations of $\operatorname{PG}$ applied to different initial sequences of integers. We study these tables of higher order differences and characterize those that have near-periodic features. As a biproduct, we also obtain two results on a class of formal power series over the field with two elements $\mathbb{F}_2$ that can be expressed as rational functions in several ways.

Key Words: Proth-Gilbreath Conjecture, quasi-periodicity, formal power series, Fibonacci sequences, SP numbers.

2020 Mathematics Subject Classification: Primary 11B37; Secondary 11B39, 11B50.

Download the paper in pdf format here.