201462: [AtCoder]ARC146 C - Even XOR
Memory Limit：1024 MB
Time Limit：2 S
Judge Style：Text Compare
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Solved：0
Description
Score : $600$ points
Problem Statement
How many sets $S$ consisting of non-negative integers between $0$ and $2^N-1$ (inclusive) satisfy the following condition? Print the count modulo $998244353$.
- Every non-empty subset $T$ of $S$ satisfies at least one of the following:
- The number of elements in $T$ is odd.
- The $\mathrm{XOR}$ of the elements in $T$ is not zero.
What is $\mathrm{XOR}$?
The bitwise $\mathrm{XOR}$ of non-negative integers $A$ and $B$, $A \oplus B$, is defined as follows:
- When $A \oplus B$ is written in base two, the digit in the $2^k$'s place ($k \geq 0$) is $1$ if exactly one of the digits in that place of $A$ and $B$ is $1$, and $0$ otherwise.
Generally, the bitwise $\mathrm{XOR}$ of $k$ non-negative integers $p_1, p_2, p_3, \dots, p_k$ is defined as $(\dots ((p_1 \oplus p_2) \oplus p_3) \oplus \dots \oplus p_k)$. We can prove that this value does not depend on the order of $p_1, p_2, p_3, \dots, p_k$.
Constraints
- $1 \le N \le 2 \times 10^5$
- All values in input are integers.
Input
Input is given from Standard Input in the following format:
$N$
Output
Print the answer.
Sample Input 1
2
Sample Output 1
15
Sets such as $\lbrace 0,2,3 \rbrace$, $\lbrace 1 \rbrace$, and $\lbrace \rbrace$ satisfy the condition.
On the other hand, $\lbrace 0,1,2,3 \rbrace$ does not.
This is because the subset $\lbrace 0,1,2,3 \rbrace$ of $\lbrace 0,1,2,3 \rbrace$ has an even number of elements, whose bitwise $\mathrm{XOR}$ is $0$.
Sample Input 2
146
Sample Output 2
743874490