Now I can draw some grid:
But what I want(I'm sorry for the weird line. I have no image processing software in my mac...):
Please notice:
f(1, 1) = f(2, 1) = f(3, 1) = 1
f(1, 1) = f(1, 2) = f(1, 3) = 1
f(2, 2) = f(1, 2) + f(2, 1) = 2
f(3, 3) = f(3, 2) + f(2, 3) = 2*(f(2, 2) + f(1, 2)) = 2*3 = 6
So there is 6 paths from (1,1) to (n,n) by two possible moves: (+1,0), (0, +1).
My question is, how to draw all paths into the grid with different color or different label?
I know the number of paths may be very large when n grows and I just want to make a nice illustration about that problem. So you can assume n <= 5.