A Fresh Power Diophantine Equation

\[ a^n-b^{n+2}=(a+b)^{n-1}\]

\(a\) 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 \(p^3-q^5=(p+q)^2.\)

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