Project Euler 140: Modified Fibonacci golden nuggets
Problem Description
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.
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.
Project Euler 140 Solution
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