For each positive integer n, consider a 2 by n rectangle. It can be tiled with n blocks, where each block is 1 by 2. How many different ways can it be tiled? What if instead of a 2 by n rectangle we were to consider a 3 by n rectangle (where now of course n would have to be an even integer). How many ways could this be tiled with 1 by 2 blocks?

Note the blocks look like [][], and can be oriented either horizontally or vertically.

Communicated by A. Kanevsky.