1.4 KiB
1.4 KiB
id | title | challengeType | forumTopicId | dashedName |
---|---|---|---|---|
5900f3fa1000cf542c50ff0c | Problem 140: Modified Fibonacci golden nuggets | 5 | 301769 | problem-140-modified-fibonacci-golden-nuggets |
--description--
Consider the infinite polynomial series A_G(x) = xG_1 + x^2G_2 + x^3G_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, \ldots
.
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 |
\frac{2}{5} |
2 |
\frac{\sqrt{22} − 2}{6} |
3 |
\frac{\sqrt{137} − 5}{14} |
4 |
\frac{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.
--hints--
modifiedGoldenNuggets()
should return 5673835352990
assert.strictEqual(modifiedGoldenNuggets(), 5673835352990);
--seed--
--seed-contents--
function modifiedGoldenNuggets() {
return true;
}
modifiedGoldenNuggets();
--solutions--
// solution required