Project Euler

Problem #137

Consider the infinite polynomial series A_(F)(x) = xF_(1) + x^(2)F_(2) + x^(3)F_(3) + ..., where F_(k) is the kth term in the Fibonacci sequence: 1, 1, 2, 3, 5, 8, ... ; that is, F_(k) = F_(k−1) + F_(k−2), F_(1) = 1 and F_(2) = 1.

For this problem we shall be interested in values of x for which A_(F)(x) is a positive integer.

Surprisingly A_(F)(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.

xA_(F)(x)
√2−11
1/22
(√13−2)/33
(√89−5)/84
(√34−3)/55

We shall call A_(F)(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.