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: $$N(r)=1+4Floor(r)+4 \sum_{x=1}^{Floor(r)}{Floor(\sqrt{r^2−x^2)}}$$ And in the case of an hexagonal lattice, I found in this post An exact counting solution for the number of points within a circle of radius $r$ centered on a lattice point in a $A_2$ hexagonal lattice 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 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.

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: $$N(r)=1+4Floor(r)+4 \sum_{x=1}^{Floor(r)}{Floor(\sqrt{r^2−x^2)}}$$ And in the case of an hexagonal lattice, I found in this post An exact counting solution for the number of points within a circle of radius $r$ centered on a lattice point in a $A_2$ hexagonal lattice 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 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. This is the structure that I obtain numerrically I am new to this subject and I appreciate all the suggestions.