PMO 2020 Problem 3

Define the sequence $\{a_i\}$ by $a_0=1$ , $a_1=4$ , and $a_{n+1}=5a_n-a_{n-1}$ for all $n\geq 1$ . Show that all terms of the sequence are of the form $c^2+3d^2$ for some integers $c$ and $d$ .