> (sqrt(2n+1)+sqrt(2n-1))^2n + (sqrt(2n+1)-sqrt(2n-1))^2n

Yes, but I gave you far more than that:

> is an integer by inspection, and (sqrt(2n+1)-sqrt(2n-1))^2n -> 0.

I identified the integer via Lucas and expanded

the tiny residuum.

Not that you wanted the full theory, of course:-)

David