Considering the number of integer solutions of $x^2+y^2\le n^2$, a digital disk, which is obviously asymptotic to $\pi n^2$, how can we find a tight upper bound of the form
$$an^2+bn+c\ ?$$
The coefficients can be real, some may be zero if that helps. This is a form of the Gauss circle problem.
Update:
From Jack's answer, we have the upper bound
$$\pi\left(n+\frac1{\sqrt2}\right)^2.$$
This is exactly the Gauss circle problem, and by Voronoi's bounds we have $$\left|\left\{ (x,y)\in\mathbb{Z}^2: x^2+y^2\leq n^2 \right\} \right| = \pi n^2+O(n^{2/3}).$$ Two slightly different proofs can be found in Section 8 of my notes.