  For any positive integer $n$ along with parameters $\alpha$ and $\nu$, we
define and investigate $\alpha$-shifted, $\nu$-offset, floor sequences of
length $n$. We find exact and asymptotic formulas for the number of integers in
such a sequence that are in a particular congruence class. As we will see,
these quantities are related to certain problems of counting lattice points
contained in regions of the plane bounded by conic sections. We give specific
examples for the number of lattice points contained in elliptical regions and
make connections to a few well-known rings of integers, including the Gaussian
integers and Eisenstein integers.
