AMO 2020 Problem 7

A tetramino tile is a tile that can be formed by gluing together four unit square tiles, edge to edge. For each natural number n, consider a bathroom whose floor is in the shape of a $2*2n$ rectangle. Let $T_n$ be the number of ways to tile this bathroom floor with tetramino tiles. Prove that, each of the numbers $T_1 , T_2 , T_3 , \cdots$ is a perfect square.