Two Kinds of Base

AtCoder

IDabc322_g

Time2000ms

Memory256MB

Difficulty—

For a non-negative integer sequence $S=(S_1,S_2,\dots,S_k)$ and an integer $a$, we define the function $f(S,a)$ as follows: * $f(S,a) = \sum_{i=1}^{k} S_i \times a^{k - i}$. For example, $f((1,2,3),4) = 1 \times 4^2 + 2 \times 4^1 + 3 \times 4^0 = 27$, and $f((1,1,1,1),10) = 1 \times 10^3 + 1 \times 10^2 + 1 \times 10^1 + 1 \times 10^0 = 1111$. You are given positive integers $N$ and $X$. Find the number, modulo $998244353$, of triples $(S,a,b)$ of a sequence of non-negative integers $S=(S_1,S_2,\dots,S_k)$ and positive integers $a$ and $b$ that satisfy all of the following conditions. * $k \ge 1$ * $a,b \le N$ * $S_1 \neq 0$ * $S_i < \min(10,a,b)(1 \le i \le k)$ * $f(S,a) - f(S,b) = X$ ## Constraints * $1 \le N \le 10^9$ * $1 \le X \le 2 \times 10^5$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $X$ [samples]

Samples

Input #1

4 2

Output #1

5

The five triples $(S,a,b)=((1,0),4,2),((1,1),4,2),((2,0),4,3),((2,1),4,3),((2,2),4,3)$ satisfy the conditions.

Input #2

9 30

Output #2

Input #3

322322322 200000

Output #3

140058961

API Response (JSON)

{
  "problem": {
    "name": "Two Kinds of Base",
    "description": {
      "content": "For a non-negative integer sequence $S=(S_1,S_2,\\dots,S_k)$ and an integer $a$, we define the function $f(S,a)$ as follows: *   $f(S,a) = \\sum_{i=1}^{k} S_i \\times a^{k - i}$. For example, $f((1,2,3",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc322_g"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "For a non-negative integer sequence $S=(S_1,S_2,\\dots,S_k)$ and an integer $a$, we define the function $f(S,a)$ as follows:\n\n*   $f(S,a) = \\sum_{i=1}^{k} S_i \\times a^{k - i}$.\n\nFor example, $f((1,2,3...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments