[math-fun] Mrs. Perkin's Quilt variation

Suppose you are asked to tile an integer square of side n with smaller squares, but the tiles are all of side 1, 2, or 3. What is the smallest number of tiles, a(n), that you can get away with? For n = 1, 2, 3, 4, 5, 6 the answers are fairly easily seen to be 1, 1, 1, 4, 8, 4; I'm not quite as sure that my value a(7) = 13 is correct. If these values are right, then a(n) is not in OEIS. It's obvious that a(3k) = k^2. I would expect that big squares can be optimally tiled by filling most of the space with 3's, with a "fringe" of some sort around two sides if n is not a multiple of 3. But I maintain a small hope that there will be "weird" solutions that don't look like this. Can anybody provide more values?