Consider a regular hexagonal lattice, like so
Given a circle centred at a vertex in this lattice, with radius $r\in\mathbb{R}^+$, is the number of lattice points inside this circle known?
Note: The purpose of this question is simply to inquire whether this is a known result, and/or to ask for suggestions on how to approach such a problem.
Some thoughts: Over a square lattice, this has been extensively studied, so I wonder if some of the techniques used there could be extended to a hexagonal lattice, where these are fundamentally different problems. This, in a way, is the hexagonal version of the Gauss Circle Problem, which was also discussed here.
Any ideas?