For any n, does there exist a polyomino with exactly n distinct domino tilings?

yes, consider the following sequence of polyominoes: +--+ | | +--+ | | +--+ +--+--+ | | | +--+--+ | | | +--+--+ +--+--+ | | | +--+--+ | | | +--+--+ | | | +--+--+ +--+--+ | | | +--+--+--+ | | | | +--+--+--+ | | | | +--+--+--+ +--+--+ | | | +--+--+--+ | | | | +--+--+--+ | | | | +--+--+--+ | | | +--+--+ +--+--+ | | | +--+--+--+ | | | | +--+--+--+--+ | | | | | +--+--+--+--+ | | | | +--+--+--+ +--+--+ | | | +--+--+--+ | | | | +--+--+--+--+ | | | | | +--+--+--+--+ | | | | +--+--+--+ | | | +--+--+ +--+--+ | | | +--+--+--+ | | | | +--+--+--+--+ | | | | | +--+--+--+--+--+ | | | | | +--+--+--+--+ | | | | +--+--+--+ ... (this sequence is surely well-known.) mike