With the help of a computer (e.g. using Dijkstra's algorithm), try plotting the knight-reachable squares for, say, $N=10,11,12,13$. You should find that you get a solid octagon (whose diameter increases linearly with $N$), surrounded by a fuzzy fringe. But the width and pattern of the fuzzy fringe is the same for each $N$. If it were getting wider and more complex, the situation would be difficult to analyze, but here we have for each $N$:
- A solid octagon, with a number of squares that grows in proportion to the square of $N-k_1$ for some $k_1$.
- Eight corner patterns which are the same for each $N$, just displaced further out.
- Fuzzy fringes, consisting of a number of identical patterns that grows in proportion to $N-k_2$ for some $k_2$.
In sum, for large $N$, the number of reachable squares ought to grow as a second-degree polynomial. Fit such a polynomial to the values for $N=10,11,12$ and verify that it works for $N=13$ too.
For $N$s that are too small to allow the central solid octagon to develop, you will probably have to override the quadratic with tabulated values. But once $N$ gets large enough, the quadratic growth will continue indefinitely.