  Given relatively prime positive integers, $a_1,\ldots,a_n$, the Frobenius
number is the largest integer with no representations of the form
$a_1x_1+\cdots+a_nx_n$ with nonnegative integers $x_i$. This classical value
has recently been generalized: given a nonnegative integer $k$, what is the
largest integer with at most $k$ such representations? Other classical values
can be generalized too: for example, how many nonnegative integers are
representable in at most $k$ ways? For sufficiently large $k$, we give formulas
for these values by understanding the level sets of the restricted partition
function (the function $f(t)$ giving the number of representations of $t$).
Furthermore, we give the full asymptotics of all of these values, as well as
reprove formulas for some special cases (such as the $n=2$ case and a certain
extremal family from the literature). Finally, we obtain the first two leading
terms of the restricted partition function as a so-called quasi-polynomial.
