Count Sequences

AtCoder

IDabc276_g

Time4000ms

Memory256MB

Difficulty—

Find the number of sequences of integers with $N$ terms, $A=(a_1,a_2,\ldots,a_N)$, that satisfy the following conditions, modulo $998244353$. * $0 \leq a_1 \leq a_2 \leq \ldots \leq a_N \leq M$. * For each $i=1,2,\ldots,N-1$, the remainder when $a_i$ is divided by $3$ is different from the remainder when $a_{i+1}$ is divided by $3$. ## Constraints * $2 \leq N \leq 10^7$ * $1 \leq M \leq 10^7$ * All values in the input are integers. ## Input The input is given from Standard Input in the following format: $N$ $M$ [samples]

Samples

Input #1

3 4

Output #1

8

Here are the eight sequences that satisfy the conditions.

*   $(0,1,2)$
*   $(0,1,3)$
*   $(0,2,3)$
*   $(0,2,4)$
*   $(1,2,3)$
*   $(1,2,4)$
*   $(1,3,4)$
*   $(2,3,4)$

Input #2

276 10000000

Output #2

909213205

Be sure to find the count modulo $998244353$.

API Response (JSON)

{
  "problem": {
    "name": "Count Sequences",
    "description": {
      "content": "Find the number of sequences of integers with $N$ terms, $A=(a_1,a_2,\\ldots,a_N)$, that satisfy the following conditions, modulo $998244353$. *   $0 \\leq a_1 \\leq a_2 \\leq \\ldots \\leq a_N \\leq M$. * ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 4000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc276_g"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Find the number of sequences of integers with $N$ terms, $A=(a_1,a_2,\\ldots,a_N)$, that satisfy the following conditions, modulo $998244353$.\n\n*   $0 \\leq a_1 \\leq a_2 \\leq \\ldots \\leq a_N \\leq M$.\n* ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments