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---
id: 5900f4b71000cf542c50ffc9
title: '問題 330歐拉數'
challengeType: 5
forumTopicId: 301988
dashedName: problem-330-eulers-number
---
# --description--
對於所有的整數 $n$,一個無限實數序列 $a(n)$ 定義如下:
$$ a(n) = \begin{cases} 1 & n < 0 \\\\
\displaystyle \sum_{i = 1}^{\infty} \frac{a(n - 1)}{i!} & n \ge 0 \end{cases} $$
例如
$$\begin{align} & a(0) = \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \ldots = e 1 \\\\
& a(1) = \frac{e 1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \ldots = 2e 3 \\\\ & a(2) = \frac{2e 3}{1!} + \frac{e 1}{2!} + \frac{1}{3!} + \ldots = \frac{7}{2} e 6 \end{align}$$
其中$e = 2.7182818\ldots$ 是歐拉常數
可以看出$a(n)$ 可以寫成 $\displaystyle\frac{A(n)e + B(n)}{n!}$ 這樣的形式其中 $A(n)$ $B(n)$ 均是整數
例如$\displaystyle a(10) = \frac{328161643e 652694486}{10!}$。
求解 $A({10}^9)$ + $B({10}^9)$ 並給出答案 $\bmod 77\\,777\\,777$。
# --hints--
`eulersNumber()` 應該返回 `15955822`
```js
assert.strictEqual(eulersNumber(), 15955822);
```
# --seed--
## --seed-contents--
```js
function eulersNumber() {
return true;
}
eulersNumber();
```
# --solutions--
```js
// solution required
```