Publish Date:
Consider the infinite polynomial series $A_G(x) = x G_1 + x^2 G_2 + x^3 G_3 + \cdots$, where $G_k$ is the $k$th term of the second order recurrence relation $G_k = G_{k-1} + G_{k-2}$, $G_1 = 1$ and $G_2 = 4$; that is, $1, 4, 5, 9, 14, 23, \dots$ .
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.
| $x$ | $A_G(x)$ |
|---|---|
| $\frac{\sqrt{5}-1}{4}$ | 1 |
| $\tfrac{2}{5}$ | 2 |
| $\frac{\sqrt{22}-2}{6}$ | 3 |
| $\frac{\sqrt{137}-5}{14}$ | 4 |
| $\tfrac{1}{2}$ | 5 |
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.
Press F12 and use the "Console" tab to view the output of your codes.