If all of their long side are vertical there is only 1 possiblity : just use $1\times2$ dominos and place it at vertical. If there long side are horizontal we will see two line of size $1\times 13$ so we just have to compute the number of possibility of covering a $1\times 13$ with only $1\times3$ and $1\times 2$ dominos. We compute that it is $20$ so the answer is $20\times20+1=401$

There are actually $16$ ways to cover a $1x13$ with only $1x3$ and $1x2$ dominoes, which can be calculated from the recurrence relationship $f(n) = f(n-2) + f(n-3)$, where $f(n)$ is the number of ways to tile a $1xn$ grid with $1x2$ and $1x3$ tiles. From here, we have $16^2+1 = \bold{257}$, which is the answer.

13 = 3 × 1 + 2 × 5 = 3 × 3 + 2 × 2 Answer = 1 + [ (1 + 5)C1 + (3 + 2)C3 ]^2 = 1 + [ 6C1 + 5C3 ]^2 = 257