MathNet 2020 — Math Olympiad Vault — Part 04 Question 141

A sequence $(a_n)_{n\ge0}$ satisfies $a_1=1$ and $a_{m+n}+a_{m-n}=\tfrac12(a_{2m}+a_{2n})$ for all $m\ge n\ge0$. Find $a_{2020}$.