I would like to count the number of sublattices of index $n $ of $\Bbb Z^2$. For $n=2$, I found three lattices : $\langle(2,0), (0,1)\rangle, \langle(0,2), (1,0)\rangle$ and $\langle(1,1), (-1,1)\rangle $. How can I find the lattices of index $n $ and how many are there?
Can we generalise for other dimensions?