# Problem 140: Modified Fibonacci golden nuggets
Consider the infinite polynomial series AG(x) = xG1 + x2G2 + x3G3 + ...,
where Gk is the kth term of the second order recurrence relation Gk =
Gk−1 + Gk−2, G1 = 1 and G2 = 4; that is, 1, 4, 5, 9, 14, 23, ... . For
this problem we shall be concerned with values of x for which AG(x) is a
positive integer. The corresponding values of x for the first five
natural numbers are shown below. xAG(x) (√5−1)/41 2/52 (√22−2)/63
(√137−5)/144 1/25 We shall call AG(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.