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Problem 140


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。