Problem 140
Modified Fibonacci golden nuggets
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.
修正斐波那契金块
考虑无穷级数AG(x) = xG1 + x2G2 + x3G3 + …，其中Gk是二阶递归关系Gk = Gk?1 + Gk?2的第k项，其中G1 = 1，G2 = 4，该序列为1, 4, 5, 9, 14, 23, … 。
在这个问题中，我们感兴趣的是那些使得AG(x)为正整数的x。
对应于前五个自然数的x如下所示。
| x | AG(x) |
|---|---|
| (√5?1)/4 | 1 |
| 2/5 | 2 |
| (√22?2)/6 | 3 |
| (√137?5)/14 | 4 |
| 1/2 | 5 |
当x是有理数时，我们称AG(x)是一个修正斐波那契金块，因为这样的数将会变得越来越稀少，例如，第20个修正斐波那契金块将是211345365。
求前30个修正斐波那契金块的和。