## Modified Fibonacci golden nuggets

### Problem 140

Published on Saturday, 3rd February 2007, 07:00 am; Solved by 2660; Difficulty rating: 55%Consider the infinite polynomial series A_{G}(*x*) = *x*G_{1} + *x*^{2}G_{2} + *x*^{3}G_{3} + ..., where G_{k} is the *k*th 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.

x | A_{G}(x) |

(√5−1)/4 | 1 |

2/5 | 2 |

(√22−2)/6 | 3 |

(√137−5)/14 | 4 |

1/2 | 5 |

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.