Problem:
Score : 800 points

Problem Statement

Given is a positive even number N.

Find the number of strings s of length N consisting of A, B, and C that satisfy the following condition:

s can be converted to the empty string by repeating the following operation:

Choose two consecutive characters in s and erase them. However, choosing AB or BA is not allowed.

For example, ABBC satisfies the condition for N=4, because we can convert it as follows: ABBC → (erase BB) → AC → (erase AC) → (empty).

The answer can be enormous, so compute the count modulo 998244353.

Constraints

2 \leq N \leq 10^7

N is an even number.

Input

Input is given from Standard Input in the following format:

N

Output

Print the number of strings that satisfy the conditions, modulo 998244353.

Sample Input 1

2

Sample Output 1

7

Except AB and BA, all possible strings satisfy the conditions.

Sample Input 2

10

Sample Output 2

50007

Sample Input 3

1000000

Sample Output 3

210055358

Solution in C++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
using namespace std;
const long long MOD=998244353;

template<uint_fast64_t Modulus> class modint{
    using u64=uint_fast64_t;
    public:
    u64 a;
    constexpr modint(const u64 x=0) noexcept:a(((x%Modulus)+Modulus)%Modulus){}
    constexpr u64 &value() noexcept{return a;}
    constexpr const u64 &value() const noexcept{return a;}
    constexpr modint &operator+=(const modint &rhs) noexcept{
        a+=rhs.a;
        if (a>=Modulus) a-=Modulus;
        return *this;
    }
    constexpr modint operator+(const modint &rhs) const noexcept{
        return modint(*this)+=rhs;
    }
    constexpr modint &operator++() noexcept{
        return ++a,*this;
    }
    constexpr modint operator++(int) noexcept{
        modint t=*this; return ++a,t;
    }
    constexpr modint &operator-=(const modint &rhs) noexcept{
        if (a<rhs.a) a+=Modulus;
        a-=rhs.a;
        return *this;
    }
    constexpr modint operator-(const modint &rhs) const noexcept{
        return modint(*this)-=rhs;
    }
    constexpr modint &operator--() noexcept{
        return --a,*this;
    }
    constexpr modint operator--(int) noexcept{
        modint t=*this; return --a,t;
    }
    constexpr modint &operator*=(const modint &rhs) noexcept{
        a=a*rhs.a%Modulus;
        return *this;
    }
    constexpr modint operator*(const modint &rhs) const noexcept{
        return modint(*this)*=rhs;
    }
    constexpr modint &operator/=(modint rhs) noexcept{
        u64 exp=Modulus-2;
        while(exp){
            if (exp&1) *this*=rhs;
            rhs*=rhs; exp>>=1;
        }
        return *this;
    }
    constexpr modint operator/(const modint &rhs) const noexcept{
        return modint(*this)/=rhs;
    }
    constexpr modint operator-() const noexcept{
        return modint(Modulus-a);
    }
    constexpr bool operator==(const modint &rhs) const noexcept{
        return a==rhs.a;
    }
    constexpr bool operator!=(const modint &rhs) const noexcept{
        return a!=rhs.a;
    }
    constexpr bool operator!() const noexcept{return !a;}
    friend constexpr modint pow(modint rhs,long long exp) noexcept{
        modint res{1};
        while(exp){
            if (exp&1) res*=rhs;
            rhs*=rhs; exp>>=1;
        }
        return res;
    }
    template<class T> friend constexpr modint operator+(T x,modint y) noexcept{
        return modint(x)+y;
    }
    template<class T> friend constexpr modint operator-(T x,modint y) noexcept{
        return modint(x)-y;
    }
    template<class T> friend constexpr modint operator*(T x,modint y) noexcept{
        return modint(x)*y;
    }
    template<class T> friend constexpr modint operator/(T x,modint y) noexcept{
        return modint(x)/y;
    }
    friend ostream &operator<<(ostream &s,const modint &rhs) noexcept{
        return s << rhs.a;
    }
    friend istream &operator>>(istream &s,modint &rhs) noexcept{
        u64 a; rhs=modint{(s >> a,a)}; return s;
    }
};

using mint=modint<MOD>;

const int MAX=1e7+10;
vector<mint> fac(MAX),finv(MAX),inv(MAX);
void COMinit(){
    fac[0]=fac[1]=1;
    finv[0]=finv[1]=1;
    inv[1]=1;
    for (int i=2;i<MAX;++i){
        fac[i]=fac[i-1]*i;
        inv[i]=-inv[MOD%i]*(MOD/i);
        finv[i]=finv[i-1]*inv[i];
    }
}
mint COM(int n,int k){
    if (n<k||n<0||k<0) return 0;
    return fac[n]*finv[k]*finv[n-k];
}

// Be careful with the value of MAX and conducting COMinit()

int main(){
    cin.tie(0);
    ios::sync_with_stdio(false);
    COMinit();
    int N; cin >> N;
    int n=N/2;
    vector<mint> dp(n+1),sum(n+1,0);
    for (int i=0;i<=n;++i) dp[i]=COM(n,i)*pow((mint)2,n-i);
    sum[n]=dp[n];
    for (int i=n-1;i>=0;--i) sum[i]=sum[i+1]+dp[i];
    mint ans=pow((mint)3,N);
    for (int i=1;i<=n;++i) ans-=dp[i]*sum[n+1-i]*2;
    cout << ans << '\n';
}