Well for n = 3...
Set l = 2:
then m can take on the range of: -2,-1,0,1,2
Set l = 1:
m = {-1,0,1}
l = 0:
m = 0
There's 9 states.
EDIT: Think of, for n = 3, every independent quantum state by labelling your state:
\psi_{n,l,m}: \psi_{3,0,0} \neq \psi_{3,1,-1} \neq...