National problem · Combinatorics

Let \(n \ge 2\) be a natural number. Maja and Peter want to colour every cell of an \(n \times n\) table either black or blue, subject to one rule: among any four cells that can be covered by a square tile of the shape drawn below - that is, in every \(2 \times 2\) block of cells - exactly two cells must be black.

Peter has already coloured the first two cells of the first row black, as shown. In how many ways can Maja colour the rest of the table?

The tile.

Peter's two black cells. The table continues to the right and downwards.