Induction · Challenge problem

How many ways are there to tile a $3 \times 8$ rectangle with $1 \times 2$ dominoes? (Break an arbitrary tiling at the first place it can be cut: $t_m = 4t_{m-1} - t_{m-2}$ for $3 \times 2m$ boards, with $t_0 = 1$, $t_1 = 3$.)