The following question is related to When are Ehrhart polynomials polynomials?. For a positive integer $n$, let $i(D,n)$ be the number of integer points in the disk $x^2+y^2\leq n^2/\pi$, so $i(D,n) = n^2 + o(n^2)$. Let $F(x)=\sum_{n\geq 0} i(D,n)x^n$. Does $i(D,n)$ satisfy any of the following progressive weaker properties? (1) $i(D,n)$ is a polynomial for $n$ sufficiently large. (2) $F(x)$ is rational. (3) $F(x)$ is algebraic. (4) $F(x)$ is D-finite. These properties seem very unlikely, but how to prove it?

We can also ask whether the error term $i(D,n)-n^2$ has similar behavior as for the disk $x^2+y^2\leq n^2$. An incidental question (quite possibly hopeless) is whether $i(D,n)=n^2$ for infinitely many $n$. The values of $n\leq 500$ for which this happens are 1, 3, 9, 11, 35, 45, 57, 61, 109, 155, 159, 227, 275, 365, 379, 383, 471, 481. (By a simple symmetry argument is is clear that $n$ must be odd.)