Given a sequence of complex numbers $(c_n)_{n\in\mathbb{N}}$, is there a nice characterization of when the Zariski closure of $\{(n,c_n):n\in \mathbb{N}\}$ in $\mathrm{A}^2_\mathbb{C}$ is not the whole plane, i.e., the set is contained in finitely many plane curves?
A related question is when the set is contained in a single curve. Clearly it is necessary that there are finitely many complex numbers (namely the coefficient of the equation defining the curve) such that if $K$ is the field generated by those numbers over $\mathbb{Q}$, then all the $c_i$'s are contained in some finite extension $K'$ of $K$.