  We consider the problem of computing $R(c,a)$, the number of unlabeled graded
lattices of rank $3$ that contain $c$ coatoms and $a$ atoms. More specifically
we do this when $c$ is fairly small, but $a$ may be large. For this task, we
describe a computational method that combines constructive listing of basic
cases and tools from enumerative combinatorics. With this method we compute the
exact values of $R(c,a)$ for $c\le 9$ and $a\le 1000$.
  We also show that, for any fixed $c$, there exists a quasipolynomial in $a$
that matches with $R(c,a)$ for all $a$ above a small value. We explicitly
determine these quasipolynomials for $c \le 7$, thus finding closed form
expressions of $R(c,a)$ for $c \le 7$.
