Problem 140

Modified Fibonacci golden nuggets

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_{k1} + G_{k2}, 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) |

(51)/4 | 1 |

2/5 | 2 |

(222)/6 | 3 |

(1375)/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.

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