Project Euler

Problem #140

Consider the infinite polynomial series A_(G)(x) = xG_(1) + x^(2)G_(2) + x^(3)G_(3) + ..., where G_(k) is the kth term of the second order recurrence relation G_(k) = G_(k−1) + G_(k−2), G_(1) = 1 and G_(2) = 4; that is, 1, 4, 5, 9, 14, 23, ... .

For this problem we shall be concerned with values of x for which A_(G)(x) is a positive integer.

The corresponding values of x for the first five natural numbers are shown below.

xA_(G)(x)
(√5−1)/41
2/52
(√22−2)/63
(√137−5)/144
1/25

We shall call A_(G)(x) a golden nugget if x is rational, because they become increasingly rarer; for example, the 20th golden nugget is 211345365.

Find the sum of the first thirty golden nuggets.