Suppose there are $n$ balls (possibly, of different sizes) in $\mathbb R^3$ such that their intersection $\mathfrak C$ is non-empty and has a positive volume (i.e. is not a single point). Apparently, $\mathfrak C$ is a convex body with a piecewise smooth surface — a "quilt" of sphere fragments. Let $f(n)$ be the maximal number of fragments that can be achieved for a given $n$.

Is there a simple formula or recurrence relation for $f(n)$?