Solution to Robot

As promised, here is a solution.

For N=15 and 9 turns, lets take a right step first. Then the path definitely looks like the one shown in the figure. Every tuple {(x1,x2,x3,x4,x5),(y1,y2,y3,y4,y5)} corresponds to a different path. Consider the equations:

x1+x2+x3+x4+x5=14 (xi>=1)-------(i)
y1+y2+y3+y4+y5=14 (yi>=1)-------(ii)

Let P be the number of solutions to (i) and Q be the number of solutions to (ii). Then the answer is 2*P*Q. The factor 2 comes from the fact that for every solution, there is a similar path where we initially take a down step.