Take an infinite hexagonal lattice (or equivalently, an equilateral triangular lattice), with unit spacing between the closest lattice point pairs, and draw a disc of radius $r$ centered on a lattice point at $(0, 0)$. Let $N(r, hex)$ denote the number of hexagonal lattice points at coordinates $(a, b)$ s.t. $(a^2 + b^2) \leq r^2$, i.e. the number of lattice points on or within the aforementioned disc of radius $r$.
Are there any literature references for approximations to $N(r, hex)$ (I haven't been able to find any through a Google search)? What is an exact counting solution for $N(r, hex)$?
Using the exact counting solution for the $Z^2$ integer lattice, (http://mathworld.wolfram.com/GausssCircleProblem.html) I suppose we can guess a lowerbound for the hexagonal lattice of:
Lowerbound $N(r, hex) = 1 + Floor[\frac{r}{2}] + 4*\sum^{Floor[\frac{r}{2}]}_{i=1} Floor[((\frac{r}{2})^2-i^2)^{\frac{1}{2}}] + 2*Floor[r]$
Where we simply overlay the $Z^2$ lattice with (closest) nearest-neighbor spacing $2$ on top of an $A_2$ hexagonal lattice with (closest) nearest-neighbor spacing $1$, and add an additional $2*Floor[r]$ correctional term.
[10/13/12] The OEIS sequences are extremely helpful, but after searching the literature for awhile, I'm still having difficulty finding an exact (counting) solution for the number of lattice points within a circle of real number radius $r$. Any references would be very much appreciated!
[10/14/12] Still no luck finding a reference in the literature. Surely someone has looked at this problem for, say, graphene and other molecular or atomic lattices where one would like to have a precise atom count a certain physical distance away from one atom?
[10/19/12] I managed to find the exact OEIS sequence I was looking for: http://oeis.org/A053416
However, I'd still like to find an exact counting solution, like the one presented above the $Z^2$ integer lattice.
Best Answer
Lax and Phillips (J. Funct. Anal. vol 46 (1982), 280--350) showed, for any crystallographic group $\Gamma$ in the Euclidean plane, that $$ N(r;x,x_0)= \frac{\pi r^2}{|F|} + O(r^{2/3} (\log r)^{1/2}), $$ as $r\to+\infty$, where $|F|$ denotes the volume of the fundamental domain of $\Gamma$, $x,x_0\in\mathbb R^2$ and $N(r;x,x_0)$ is the number of elements $\gamma\in\Gamma$ such that $$ |x-\gamma (x_0)|\leq r. $$
Later, Levitan (Russian Math. Surveys vol 42:3 (1987), 13--42) improved the error term to $O(r^{2/3})$.
In your particular case, $\Gamma$ is the subgroup of the isometries of the plane generated by the translations for $(1,0)$ and $(1/2,\sqrt{3}/2)$, thus $|F|=\sqrt{3}/2$.
Both papers works in higher dimensions and in hyperbolic spaces.