Consider $f(n)$, defined as the number of unique quotients, including the three undefined values when applicable ($\frac{1}{0}$, $\frac{-1}{0}$ and $\frac{0}{0}$), of $p$ and $q$ given that $p$ and $q$ are drawn from the integers between $-n$ and $n$ inclusive as follows:
$$g(n) = \{\frac{p}{q} \ | \ p,q \in [-n..n]\} \\ f(n) = \textrm{Number of unique values in} \ g(n)$$
This rather simple function, when applied to the integers from 1 to 20 yields the sequence $$[6,10,18,26,42,50,74,90,114,130,170,186,234,258,290,322,386,410,482,514, \dots]$$
If you exclude the three undefined values, and as such subtract $3$ from each element in the sequence, you end up with
$$[3,7,15,23,39,47,71,87,111,127,167,183,231,255,287,319,383,407,479,511,\dots]$$
According to the Online Encyclopedia of Integer Sequences, this is the same exact sequence as the number of 2 x 2 integer matrices with entries defined from $\{0, 1, \dots n\}$ having determinant $1$.
Why is that?