Patio Tiling Proof

Question

Call a square grid n squares wide, with a single square removed from the top left, a patio. It appears that the minimum number of squares necessary to tile a patio 2^k squares wide is 3k. This is done by adding 3 larger squares to the bottom right of the patio, each time the patio doubles in width. Can 3k be proven minimal? Here are some patios of width n = 4 through 8, in case my definition of patios was unclear.

Answer 1

You can show inductively that $3k$ is an upper bound on the minimum, and that bound is tight for $k\in\{1,2,3,4\}$. But for $k=5$, the minimum is $14$: