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?




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[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.



The Gauss circle problem on a hexagonal lattice
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.


