Sugoroku 3

AtCoder

IDabc263_e

Time2000ms

Memory256MB

Difficulty—

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. ## 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: $N$ $A_1$ $A_2$ $\dots$ $A_{N-1}$ [samples] ## 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 $R$.

Samples

Input #1

3
1 1

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 $\frac{1}{8}$.

Input #2

5
3 1 2 1

Output #2

332748122

API Response (JSON)

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  "problem": {
    "name": "Sugoroku 3",
    "description": {
      "content": "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 ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc263_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There are $N$ squares called Square $1$ though Square $N$. You start on Square $1$.\nEach of the squares from Square $1$ through Square $N-1$ has a die on it. The die on Square $i$ is labeled with the ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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