2005 AIME I Problem 13

A particle moves in the Cartesian plane from one lattice point to another according to the following rules:

1. From any lattice point (a, b), the particle may move only to (a+1, b),(a, b+1), or (a+1, b+1).
2. There are no right angle turns in the particle’s path. That is, the sequence of points visited contains neither a subsequence of the form (a, b),(a+1, b),(a+ 1, b+1 ) nor a subsequence of the form (a, b),(a, b+1),(a+1, b+1).

How many different paths can the particle take from (0,0) to (5,5) ?