A002805

A002805 is a sequence in the On-Line Encyclopedia of Integer Sequences (OEIS). The sequence is defined as the number of ways to tile a 3xn rectangle with 1x2 dominoes. This is a combinatorial problem that has been studied in mathematics. The initial terms of the sequence are 0, 3, 0, 11, 0, 41, 0, 153, 0, 571, ...

The problem of tiling a 3xn rectangle with dominoes has a connection to the study of perfect matchings in grid graphs. For an n-by-m grid, a perfect matching is a set of edges such that every vertex is incident to exactly one edge in the set. In the context of domino tiling, each domino covers two adjacent squares, which corresponds to an edge between the centers of those squares in the grid graph.

The recurrence relation for the sequence A002805 can be derived by considering how the rightmost column(s) of the 3xn rectangle can be tiled. However, it is known that for odd n, the number of tilings is zero. This is because each domino covers two squares, so any tiled area must have an even number of squares. A 3xn rectangle has 3n squares, which is odd if n is odd, making tiling impossible. For even n, the number of tilings follows a linear recurrence relation. The generating function for the sequence has also been determined.