Is there a general formula for number of integral points inside the circle $x^2+y^2=a^2$ for $a \in \mathbb Z^+$

Question

I could work out a general formula for number of integral (lattice) points lying on or inside the circle $x^2+y^2=a^2$, $a \in\mathbb{Z}^+$

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I tried to work out a general formula for number of integral (lattice) points lying strictly inside the circle $x^2+y^2=a^2$, $a \in\mathbb{Z}^+$. Well, I can remove points lying on the circle when in my formula term inside the greatest integer is an integer, but i was wondering if there is any general formula possible for the same.

Answer 1

I don't think there's a neat formula for it.

This is called the Gauss circle problem. You can find more details here https://en.m.wikipedia.org/wiki/Gauss_circle_problem.