Let
![$\operatorname{PG}$](img1.png)
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
![$\operatorname{PG}$](img1.png)
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
![$\mathbb{F}_2$](img2.png)
that
can be expressed as rational functions in several ways.