Question

Let S be the set of {(1,1), (1,−1), (−1,1), (1,0), (0,1)}-lattice paths which begin at (1,1), do not use the same vertex twice, and never touch either the x-axis or the y-axis. Let Sx,y be the set of paths in S which end at (x,y).For how many ordered pairs (x,y) subject to 1≤x,y≤31, is |Sx,y| a multiple of 3?
Details and assumptions
A lattice path is a path in the Cartesian plane between points with integer coordinates.
A step in a lattice path is a single move from one point with integer coordinates to another.
The size of the step from (x1,y1) to (x2,y2) is (x2−x1,y2−y1).
The length of a lattice path is the number of steps in the path.
For a set S={(xi,yi)}ki=1, an S-lattice path is a lattice path where every step has size which is a member of S.