This Mathematica formula reproduces the numbers in the OEIS and should be self explanatory:
n[r_] := Sum[ 1 + 2 Floor[Sqrt[r^2 - 3 x^2]], {x, -Floor[r/Sqrt[3]], Floor[r/Sqrt[3]]}] + Sum[2 Floor[ Sqrt[r^2 - 3 x^2] + 1/2], {x, -Floor[(r/Sqrt[3]) + 1/2] + 1/2, Floor[(r/Sqrt[3]) + 1/2] - 1/2}]
http://oeis.org/A053416 has the values corresponding to n[1/2], n[1], n[3/2], n[2] etc.
Assuming the lattice is generated by $(0,1)$ and $(\sqrt{3}/2,1/2)$, then the first sum counts the number of points of the form $(\sqrt{3}x,y)$ where $x,y\in\mathbb{Z}$, $\sqrt{3}x\le r$, and $3x^2+y^2\le r^2$. The second sum counts the number of points of the form $(\sqrt{3}x,y)$ where $x,y\in(\mathbb{Z}+1/2)$, $\sqrt{3}x\le r$, and $3x^2+y^2\le r^2$.
