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Unique Hyperbola

y2xyx2=1y^2 - xy - x^2 = 1

Let (x,y)(x,y) be the non-negative integer solutions to the hyperbolic graph above.

If x+y=nx+y = n for some perfect square nn, what is the sum of all possible n?n?

Hint: The only Fibonacci numbers that are perfect squares are 0, 1, and 144.


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