1.3 KiB
1.3 KiB
id | title | challengeType | forumTopicId | dashedName |
---|---|---|---|---|
5900f4b71000cf542c50ffc9 | Problem 330: Euler's Number | 5 | 301988 | problem-330-eulers-number |
--description--
An infinite sequence of real numbers a(n)
is defined for all integers n
as follows:
a(n) =
\begin{cases}
1 & n < 0 \\\\
\displaystyle \sum_{i = 1}^{\infty} \frac{a(n - 1)}{i!} & n \ge 0
\end{cases}
For example,
\begin{align}
& a(0) = \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \ldots = e − 1 \\\\
& a(1) = \frac{e − 1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \ldots = 2e − 3 \\\\
& a(2) = \frac{2e − 3}{1!} + \frac{e − 1}{2!} + \frac{1}{3!} + \ldots = \frac{7}{2} e − 6
\end{align}$$
with $e = 2.7182818\ldots$ being Euler's constant.
It can be shown that $a(n)$ is of the form $\displaystyle\frac{A(n)e + B(n)}{n!}$ for integers $A(n)$ and $B(n)$.
For example $\displaystyle a(10) = \frac{328161643e − 652694486}{10!}$.
Find $A({10}^9)$ + $B({10}^9)$ and give your answer $\bmod 77\\,777\\,777$.
# --hints--
`eulersNumber()` should return `15955822`.
```js
assert.strictEqual(eulersNumber(), 15955822);
```
# --seed--
## --seed-contents--
```js
function eulersNumber() {
return true;
}
eulersNumber();
```
# --solutions--
```js
// solution required
```