GCP (Gauss' Circle Problem) asks for a closed form for the number of square-lattice points inside a circle, centered at the origin, of radius $r$.

Let's denote by $N(r)$ the number of these points. Then, $N(r)$ is the number of integer solutions (pairs of integers $x$ and $y$) to the inequality

$$x^2+y^2 \le r^2$$

But, what would happen if, instead of setting the center of the circle at the origin, we moved the circle $1/2$ units in the X-axis? The number of lattice points $N^*(r)$ would be the number of integer solutions to

$$(x+1/2)^2+y^2 \le r^2$$

It is easy to show that $N^*(r)$ would also be the number of solutions to

$$x^2+(y+1/2)^2 \le r^2$$

For last, let's denote by $N^{**}(r)$ the number of lattice points of a circle centered at $(1/2, 1/2)$; that is, the number of integer solutions to

$$(x+1/2)^2+(y+1/2)^2 \le r^2$$

Then, my main question is:

*Is it possible to get either $N^*(r)$ or $N^{**}(r)$ if only $N(r)$ and $r$ are given?*

I found this post about a generalization of this problem, but I would prefer having an exact answer to this question, rather than an approximate result involving big-O notation. It does not mind if any floor function is needed.

Thank you.