## Modified Fibonacci golden nuggets

### Problem 140

Published on Saturday, 3rd February 2007, 07:00 am; Solved by 2035

Consider the infinite polynomial series AG(x) = xG1 + x2G2 + x3G3 + ..., where Gk is the kth term of the second order recurrence relation Gk = Gk1 + Gk2, 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.

x | AG(x)
(51)/4 | 1
2/5 | 2
(222)/6 | 3
(1375)/14 | 4
1/2 | 5

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.