<s>[INST] 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
 [/INST] 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)
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