Consider a minimum-distance packing of unit circles (aka pennies) that form a hexagonal tiling. If we restrict our attention to only those pennies that are contained or tangent to a concentric circle of integer diameter $2k+1$, we capture $3k(k+1)+1$ pennies:
for $k\in\mathbb{N}$.
Question
Does there exist a new arrangement of those $3k(k+1)+1$ pennies wherein the pennies contain a non-zero uniform distance between neighbors? Two pennies are neighbors if their centers are within $3$ of each other (feel free to refine this definition). Bonus points for showing how to construct such an arrangement, if it is possible.
the naive example:
We might choose to naively construct such an arrangement by taking the $6k$ perimeter pennies and evenly distributing them over a circle of diameter $2k$. Repeating this for all $k-1$ lower layers yields the example arrangement:
We see this introduced uniform spacing between pennies on a common layer, but this spacing is not uniform amongst pennies from adjacent layers.
Given a spaced-out packing inside the large circle of radius $9$ where all sufficiently-close circles have centers at a distance $d>2$ from each other, we could replace each disc with a disc of radius $d/2$, which would form a packing of larger discs that did touch each other inside a large circle of radius $8+d/2$. Rescaling, that corresponds to packing a unit circle with discs of radius
$$\frac{d/2}{8+d/2} = \frac{d}{16+d}>\frac19$$
Conversely, given a packing of the unit circle with discs of radius $r>1/9$, there will be some minimum distance $x>2r$ between non-touching circles (since there are only finitely many distances to check), and so there will be some threshold $t=\frac{8x}{1-r}>2$ such that any discs within distance less than $t$ are at the same distance from each other. (This doesn't guarantee we'll have $t\ge 3$ to meet the definition in the original post, though.)
Conveniently, in the framing of "finding dense packings of a unit disc with equally-sized discs", there is some existing work on the problem, though proving optimality of configurations is notoriously hard. The best-known configuration with $61$ discs looks like this:
It offers a radius of $r=0.11545614167835687\ldots>\frac19$, which corresponds to a packing of unit discs in a circle of radius $9$ which are separated uniformly by a distance of $16r/(1-r)\approx2.0884$. Downloading the coordinates and measuring the minimum non-touching distance, we find $x\approx0.28114$ and hence $t\approx 2.5427$.
Scaling down the circles to be unit circles in a disk of radius $9$ looks like this:
Highlighted in blue and green are the distances between circles of length $2.54$ and $2.95$ respectively, the only two distances less than $3$ between non-tangent circles in this arrangment.
I would expect finding a packing in which the minimum distance between non-tangent circles is so high to be quite difficult, and weakly guess that it is impossible.