On Quasi-Periodicity in Proth-Gilbreath Triangles

TL;DR

This paper investigates the quasi-periodic patterns in the infinite tables generated by iteratively applying the Proth-Gilbreath operator to integer sequences, revealing structural properties and connections to rational functions over F2.

Contribution

It characterizes near-periodic features in higher order difference tables and links these patterns to rational functions over the field with two elements.

Findings

Identification of conditions for near-periodicity in difference tables

Characterization of certain formal power series over F2 as rational functions

New insights into the structure of Proth-Gilbreath iterates

Abstract

Let 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 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 F2 that can be expressed as rational functions in several ways.