My first plan of attack would be to find a formula for the nth term as a function of n directly, without making reference to the Fibonacci sequence. Presumably, it will be a polynomial in phi.
= a_(n-1) + a_(n-2) (n greater than? or equal to 2), phi the golden ratio (1+sqrt(5))/2. Find the sum of the series sum for n = 0 to infinity of (a_n * phi - a_(n+1))^2.