 Frobenius trace recurrence

For an elliptic curve over ${\mathbb{F}}_q$, the Frobenius isogeny has characteristic polynomial $T^2-tT+q$. Taking traces of its powers gives this recurrence. If $\alpha ,\beta$ are the roots, then $t_n={\alpha }^n+{\beta }^n$ and $\#E({\mathbb{F}}_{q^n})=q^n+1-t_n$. The Hasse bound implies $|\alpha |=|\beta |=\sqrt{q}$, including the repeated-root case.