How many different knight's moves are there on an $n \times n$ chessboard? I have found there to be $48$ total moves on a $4 \times 4$ board... and $96$ on a $5 \times5$... but I can not see the relevance to each other in terms of a "$n  \times n$" board. 
By "moves" I am referring to from each space the amount of moves available from it.
Example

Extending your example ... 
For the top row you get 2,3,4,4,....,4,4,3,2
For the second row you get 3,4,6,6,....,6,6,4,3
For the third 4,6,8,8,...,8,8,6,4
For the fourth also 4,6,8,8,...,8,8,6,4
...
[and then the mirror of that for the bottom 4 rows]
So:
With $n\ge 4$ you get:
$2+3+(n-4)*4+3+2=10 +4*(n-4)$ for rows 1 and $n$
$3+4+(n-4)*6+4+3=14+6*(n-4)$ for rows 2 and $n$
$(n-4)*(4+6+(n-4)*8+6+4=(n-4)*(20+8*(n-4))$ for all other rows
For a total of 
$$2*(10+14)+ (2*(4+6)+20)*(n-4)+ 8*(n-4)^2 = $$
$$48+40*(n-4)+8*(n-4)^2$$
Check: for $n=4$ we indeed get 48. For $n=5$, we get $48+40+8=96$ (So your 98 was a bit off)
