A Fresh Power Diophantine Equation

Number Theory Level 4

$a^n-b^{n+2}=(a+b)^{n-1}$

$a$ and $b$ are relatively prime positive integers and $n\, (>1)$ is an integer. Find all solutions to the equation above and enter your answer as $\sum (a+b+n).$

This is a generalization of the problem from 1997 Russian Olympiad:

For prime numbers $p$ and $q,$ solve $p^3-q^5=(p+q)^2.$