Modified Fibonacci golden nuggets
Problem 140
Consider the infinite polynomial series , where is the th term of the second order recurrence relation , and ; that is, .
For this problem we shall be concerned with values of for which is a positive integer.
The corresponding values of for the first five natural numbers are shown below.
| 1 | |
| 2 | |
| 3 | |
| 4 | |
| 5 |
We shall call a golden nugget if is rational, because they become increasingly rarer; for example, the 20th golden nugget is 211345365.
Find the sum of the first thirty golden nuggets.