If you wonder why the function is a quadratic, I suggest drawing stuff on a piece of paper. Essentially, if there were no obstacles, the furthest reachable cells would form a large diamond, which is tiled by some copies of the diamond in the input and some copies of the corners. As these have constant size, and the large diamond will grow quadratically with steps, you need a quadratic number of copies (by drawing, you can see that if steps = k * width + width/2, then there are floor((2k + 1)^2/2) copies of the center diamond, and ceil((2k + 1)^2/2) copies of each corner around).

What complicates this somewhat is that you don't just have to be able to reach a square in the number of steps, but that the parity has to match: By a chessboard argument, you can see any given square only every second step, as each step you move from a black tile to a white one or vice versa. And the parities flip each time you cross a boundary, as the input width is odd. So actually you have to either just guess the coefficients of a quadratic, as you and @hades@lemm.ee did, or do some more working out by hand, which will give you the explicit form, which I did and can't really recommend.