Modified Fibonacci golden nuggets
February 03, 2007
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.
| x | AG(x) |
|---|---|
| (√5−1)/4 | 1 |
| 2/5 | 2 |
| (√22−2)/6 | 3 |
| (√137−5)/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.
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