Unique Hyperbola

\[y^2 - xy - x^2 = 1\]

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

If \(x+y = n\) for some perfect square \(n\), what is the sum of all possible \(n?\)

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

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