Suppose there are $n$ spheres that we label $i = 1, \ldots, n$. Then suppose that the center $p_i$ of each of these spheres cannot be within distance $r$ of any other sphere. I would like to find out some information related to the most efficient way of packing these spheres.
Specifically, let us denote a sphere of radius $r$ at position $p$ by $S(p,r)$. Are there references available, or known sequences, that describe the following quantity?
$$V_{n} = \min_{p_1, \ldots, p_n} \left\{ \operatorname{volume} \left(\bigcup_{i=1}^n S(p_i,r)\right) \mid \operatorname{distance}(p_i,p_j) \geq r\ \forall i,j \right\}$$
Of particular interest is the two-dimensional case.
Some material on Sphere Packing with Overlap:
http://www.carolineuhler.com/Uhler_sphere_packing.pdf
http://arxiv.org/pdf/1401.0468v1.pdf