2005 AIME I Problem 13

A particle moves in the Cartesian Plane according to the following rules:

- From any lattice point (a,b), the particle may only move to (a+1,b), (a,b+1), or (a+1,b+1).

- There are no right angle turns in the particle's path.

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

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.