# Problem 137: Fibonacci golden nuggets
Consider the infinite polynomial series AF(x) = xF1 + x2F2 + x3F3 + ...,
where Fk is the kth term in the Fibonacci sequence: 1, 1, 2, 3, 5, 8,
... ; that is, Fk = Fk−1 + Fk−2, F1 = 1 and F2 = 1. For this problem we
shall be interested in values of x for which AF(x) is a positive
integer. Surprisingly AF(1/2)  =  (1/2).1 + (1/2)2.1 + (1/2)3.2 +
(1/2)4.3 + (1/2)5.5 + ...    =  1/2 + 1/4 + 2/8 + 3/16 + 5/32 + ...  
 =  2 The corresponding values of x for the first five natural numbers
are shown below. xAF(x) √2−11 1/22 (√13−2)/33 (√89−5)/84 (√34−3)/55 We
shall call AF(x) a golden nugget if x is rational, because they become
increasingly rarer; for example, the 10th golden nugget is 74049690.
Find the 15th golden nugget.