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