Minimum possible distance between $n$ grid points

Question

We are given a grid and a set $S$ of $n$ points on it (i.e points in the plane with integer coordinates).
We define the diamatar $diam(S)$ of $S$ to be the maximum possible distance between two points in $S$, that is, $diam(S) = \max\{distance(a, b):a,b \in S \}$.
What is the minimal possible value of $diam(S)$ for general $n$? Is there a closed formula, in terms of $n$?
For example, for $n=1$ the answer is $0$. For $n=2$ the answer is $1$ and for $n=5$ the answer is $2$.

I would also like to know closed formulas for special cases, for example, for the case where $n$ is a perfect sqaure.

Answer 1

A circle is clearly a sensible shape to consider for a 'close-packing' of integer points. For a circle of given radius $r$, (diameter $2r$), the numbers of points can easily be calculated. The solutions for small integer values of $r$ are given in A000328 - OEIS.

For example the value of $5$, as you've already determined, is given for $r=1$ and $13$ is the value for $r=2$.

If you've not already found it, https://en.wikipedia.org/wiki/Gauss_circle_problem is also a site you should visit. In particular it gives an approximate relation between the diameter, $d$, and $n$.

$$\pi d^2\approx 4n.$$

As you will see the general problem is difficult, without precise formulae.