On quasi-periodicity in Proth-Gilbreath triangles

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 F-2 that can be expressed as rational functions in several ways.