Score : $500$ points

问题陈述

有 $N$ 个正方形，分别称为第 1 个正方形到第 $N$ 个正方形。你从第 1 个正方形开始。

从第 1 个正方形到第 $N-1$ 个正方形上的每个正方形都有一颗骰子。在第 $i$ 个正方形上的骰子标记着从 $0$ 到 $A_i$ 的整数，且每个数字出现的概率相等。（各骰子的滚动是相互独立的。）

直到你到达第 $N$ 个正方形之前，你需要重复在当前所在的正方形上投掷骰子。这里的规则是：如果在第 $x$ 个正方形上投掷骰子得到整数 $y$，则你将移动到第 $x+y$ 个正方形。

求模 $998244353$ 后，投掷骰子次数的期望值。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

There are $N$ squares called Square $1$ though Square $N$. You start on Square $1$.

Each of the squares from Square $1$ through Square $N-1$ has a die on it. The die on Square $i$ is labeled with the integers from $0$ through $A_i$, each occurring with equal probability. (Die rolls are independent of each other.)

Until you reach Square $N$, you will repeat rolling a die on the square you are on. Here, if the die on Square $x$ rolls the integer $y$, you go to Square $x+y$.

Find the expected value, modulo $998244353$, of the number of times you roll a die.

Notes

It can be proved that the sought expected value is always a rational number. Additionally, if that value is represented $\frac{P}{Q}$ using two coprime integers $P$ and $Q$, there is a unique integer $R$ such that $R \times Q \equiv P\pmod{998244353}$ and $0 \leq R \lt 998244353$. Find this $R$.

Constraints

• $2 \le N \le 2 \times 10^5$
• $1 \le A_i \le N-i(1 \le i \le N-1)$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$

$A_1$ $A_2$ $\dots$ $A_{N-1}$

Output

Print the answer.

Sample Input 1

3
1 1

Sample Output 1

4

The sought expected value is $4$, so $4$ should be printed.

Here is one possible scenario until reaching Square $N$:

• Roll $1$ on Square $1$, and go to Square $2$.
• Roll $0$ on Square $2$, and stay there.
• Roll $1$ on Square $2$, and go to Square $3$.

This scenario occurs with probability $\frac{1}{8}$.

Sample Input 2

5
3 1 2 1

Sample Output 2

332748122

update @ 2024/3/10 11:08:00