For n=2, there are exactly 3 ways: all horizontal in each row, two vertical pairs, or a mix. For n=4, you get 11 ways. These numbers hint at a pattern.
Notice that n must be even. With an odd n, you have 3n cells (odd total), but each domino covers 2 cells. You can't tile an odd number of cells with dominoes. The sequence 3,11,41,153,… follows a pattern where each term depends on previous terms. Can you see it? Each new term relates to previous terms in a specific way.