In the case of a square lattice, the exact number of points within a circle of radius r centered in the center is (see: [http://mathworld.wolfram.com/GausssCircleProblem.html](https://mathworld.wolfram.com/GausssCircleProblem.html):
$$N(r)=1+4Floor(r)+4 \sum_{x=1}^{Floor(r)}{Floor(\sqrt{r^2−x^2)}}$$
And in the case of an hexagonal lattice, I found in this post [https://mathoverflow.net/questions/110186/an-exact-counting-solution-for-the-number-of-points-within-a-circle-of-radius-r](https://mathoverflow.net/questions/110186/an-exact-counting-solution-for-the-number-of-points-within-a-circle-of-radius-r) that the number of points within a circle of radius r centered in the center is:
$$ N(r)= \sum_{x = -Floor(\frac{r}{\sqrt{3}})}^{Floor(\frac{r}{\sqrt{3}})}( 1 + 2 Floor(\sqrt{r^2 - 3 x^2}) +  \sum_{x = -Floor(\frac{r}{\sqrt{3}} + \frac{1}{2}) + \frac{1}{2}}^{Floor(\frac{r}{\sqrt{3}} + \frac{1}{2}) - \frac{1}{2}}(  2 Floor(\sqrt{r^2 - 3 x^2} + \frac{1}{2}). $$ 
And I checked this expression with the values in [http://oeis.org/A053416](http://oeis.org/A053416) and I don't obtain the same values for an r.
Can you guide me where I am wrong?
In my research I want to obtain the number of the lattice points for square and hexagonal lattices in function of lattice constant and region size.

I am new to this subject and I appreciate all the suggestions.