Densest safe disk packing

Question

Inspired by current regulations regarding the minimal distance to be kept among people to prevent spreading of the COVID-19 virus and the maximal number of people in a group that is not subjected to that distance-regulation, I'd like to ask the following:

given

• a minimal distance $R\ \gt\ 0$ to be kept between individuals in a "group"
• a maximal number $k$ of individuals that are not subjected to the minimal-distance rule for groups, i.e. for each individual there are maximally $k-1$ others that are closer than $R$.

Modeling the individuals as unit circles leads to the

Question:
how many unit disks can be maximally packed in a square with sidelength $a=n*R$ under the constraint that for each disk the distance of the $k$-th nearest disk is at least $R$.

Answer 1

This is an initial idea for an approximate solution for finding the densest safe packing of an infinite set of disks in the euclidean plane:

• determine the smallest regular hexagon in which $k$ unit circles can be packed, for small $k$ known solutions can be found here and let $\varrho$ denote the radius of that hexagon's incircle

• place copies of that hexagon at each vertex of a triangular grid with edge-lengths equal to $2\varrho+R$ and rotated to render the hexagon's sides orthogonal to the grid's edges.

• exploit potential for further improvements by convoluting the unit disks inside the hexagons with open disks of radius $\frac{R}{2}$ and checking whether the distance of gridlines can be reduced without generating an overlap of convoluted circles in different hexagons.