Keep it Simple
$$ \frac{1}{n} = \frac{1}{n+1}+\frac{1}{n(n+1)}$$
works for any fraction.
To find out how many ways a unit fraction can be written as the sum
of two unit fractions, consider:
$$ \frac{1}{n} = \frac{1}{n+k}+\frac{k}{n(n+k)}$$
This is true for any value of $k$, but it only gives two unit
fractions if $k$ is a factor of $n(n+k)$
$n(n+k) = n^2+kn$ so as $k$ is always a factor of $kn$, $k$ must be
a factor of $n^2$.
I can find all the unit fractions by finding all the factors of
$n^2$ and using these as values for $k$ in the above
expression.
Example: $n=30$, so $n^2=900$.
One factor pair is $5$ and $180$ so I can take $k=5$ to give the
fraction $\frac{1}{35}$ and I can take $k=180$ to give the fraction
$\frac{1}{210}$.
$$\frac{1}{30}=\frac{1}{35}+\frac{1}{210}$$
All the other unit fraction pairs can be generated in the same way
from the other factor pairs of $900$.