(In reply to 'generated'
re Jer Your solution would include all series with positive integers each below 520, and a<b. They would include inter alia 520 as a number generated by you, contradicting the text.
re. Steve. a=b generates a regular Fibonacci series (a=b=1) ,each number being multiplied by 520/f f being a Fibonacci number dividing 520 (520/2,520/5,520/8,520/13). I just excluded them.
Both modifications would fit an altered text, creating another valid puzzle.
re: All solvers-
1. One does have to list all solutions, a counting is possible like
for u(3 )=520 there are 259 solutions: u(1)=260-k, u(2)=260+k, etc
2. The 1st part of the puzzle is still open.