Score : $600$ points

问题陈述

找出满足以下条件的整数序列 $A=(a_1,a_2,\ldots,a_N)$ 的个数，对 $998244353$ 取模。

• $0 \leq a_1 \leq a_2 \leq \ldots \leq a_N \leq M$。
• 对于每个 $i=1,2,\ldots,N-1$，当 $a_i$ 除以 $3$ 的余数与 $a_{i+1}$ 除以 $3$ 的余数不相同时。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

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:

$N$ $M$

Output

Print the answer.

Sample Input 1

3 4

Sample Output 1

8

Here are the eight sequences that satisfy the conditions.

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

Sample Input 2

276 10000000

Sample Output 2

909213205

Be sure to find the count modulo $998244353$.

update @ 2024/3/10 11:38:29