結果

問題 No.3182 recurrence relation’s intersection sum
ユーザー 👑 Nachia
提出日時 2025-06-13 21:51:38
言語 C++17
(gcc 13.3.0 + boost 1.87.0)
結果
AC  
実行時間 2 ms / 2,000 ms
コード長 7,156 bytes
コンパイル時間 1,228 ms
コンパイル使用メモリ 87,180 KB
実行使用メモリ 7,844 KB
最終ジャッジ日時 2025-06-13 21:51:41
合計ジャッジ時間 2,554 ms
ジャッジサーバーID
（参考情報） 		judge5 / judge2
このコードへのチャレンジ
（要ログイン）
ファイルパターン 結果
sample AC * 3
other AC * 40
権限があれば一括ダウンロードができます

ソースコード

diff # 

#ifdef NACHIA
#define _GLIBCXX_DEBUG
#else
#define NDEBUG
#endif
#include <iostream>
#include <string>
#include <vector>
#include <algorithm>
using i64 = long long;
using u64 = unsigned long long;
#define rep(i,n) for(i64 i=0; i<i64(n); i++)
const i64 INF = 1001001001001001001;
template<typename A> void chmin(A& l, const A& r){ if(r < l) l = r; }
template<typename A> void chmax(A& l, const A& r){ if(l < r) l = r; }
using namespace std;

#include <utility>

#include <cassert>
namespace nachia{

// ax + by = gcd(a,b)
// return ( x, - )
std::pair<long long, long long> ExtGcd(long long a, long long b){
    long long x = 1, y = 0;
    while(b){
        long long u = a / b;
        std::swap(a-=b*u, b);
        std::swap(x-=y*u, y);
    }
    return std::make_pair(x, a);
}

} // namespace nachia

namespace nachia{

template<unsigned int MOD>
struct StaticModint{
private:
    using u64 = unsigned long long;
    unsigned int x;
public:

    using my_type = StaticModint;
    template< class Elem >
    static Elem safe_mod(Elem x){
        if(x < 0){
            if(0 <= x+MOD) return x + MOD;
            return MOD - ((-(x+MOD)-1) % MOD + 1);
        }
        return x % MOD;
    }

    StaticModint() : x(0){}
    StaticModint(const my_type& a) : x(a.x){}
    StaticModint& operator=(const my_type&) = default;
    template< class Elem >
    StaticModint(Elem v) : x(safe_mod(v)){}
    unsigned int operator*() const { return x; }
    my_type& operator+=(const my_type& r) { auto t = x + r.x; if(t >= MOD) t -= MOD; x = t; return *this; }
    my_type operator+(const my_type& r) const { my_type res = *this; return res += r; }
    my_type& operator-=(const my_type& r) { auto t = x + MOD - r.x; if(t >= MOD) t -= MOD; x = t; return *this; }
    my_type operator-(const my_type& r) const { my_type res = *this; return res -= r; }
    my_type operator-() const noexcept { my_type res = *this; res.x = ((res.x == 0) ? 0 : (MOD - res.x)); return res; }
    my_type& operator*=(const my_type& r){ x = (u64)x * r.x % MOD; return *this; }
    my_type operator*(const my_type& r) const { my_type res = *this; return res *= r; }
    bool operator==(const my_type& r) const { return x == r.x; }
    my_type pow(unsigned long long i) const {
        my_type a = *this, res = 1;
        while(i){ if(i & 1){ res *= a; } a *= a; i >>= 1; }
        return res;
    }
    my_type inv() const { return my_type(ExtGcd(x, MOD).first); }
    unsigned int val() const { return x; }
    int hval() const { return int(x > MOD/2 ? x - MOD : x); }
    static constexpr unsigned int mod() { return MOD; }
    static my_type raw(unsigned int val) { auto res = my_type(); res.x = val; return res; }
    my_type& operator/=(const my_type& r){ return operator*=(r.inv()); }
    my_type operator/(const my_type& r) const { return operator*(r.inv()); }
};

} // namespace nachia
using Modint = nachia::StaticModint<998244353>;

namespace nachia{

template<class Modint>
class Comb{
private:
    std::vector<Modint> F;
    std::vector<Modint> iF;
public:
    void extend(int newN){
        int prevN = (int)F.size() - 1;
        if(prevN >= newN) return;
        F.resize(newN+1);
        iF.resize(newN+1);
        for(int i=prevN+1; i<=newN; i++) F[i] = F[i-1] * Modint::raw(i);
        iF[newN] = F[newN].inv();
        for(int i=newN; i>prevN; i--) iF[i-1] = iF[i] * Modint::raw(i);
    }
    Comb(int n = 1){
        F.assign(2, Modint(1));
        iF.assign(2, Modint(1));
        extend(n);
    }
    Modint factorial(int n) const { return F[n]; }
    Modint invFactorial(int n) const { return iF[n]; }
    Modint invOf(int n) const { return iF[n] * F[n-1]; }
    Modint comb(int n, int r) const {
        if(n < 0 || n < r || r < 0) return Modint(0);
        return F[n] * iF[r] * iF[n-r];
    }
    Modint invComb(int n, int r) const {
        if(n < 0 || n < r || r < 0) return Modint(0);
        return iF[n] * F[r] * F[n-r];
    }
    Modint perm(int n, int r) const {
        if(n < 0 || n < r || r < 0) return Modint(0);
        return F[n] * iF[n-r];
    }
    Modint invPerm(int n, int r) const {
        if(n < 0 || n < r || r < 0) return Modint(0);
        return iF[n] * F[n-r];
    }
    Modint operator()(int n, int r) const { return comb(n,r); }
    Modint parityToSign(long long x) const {
        return Modint(x%2 == 0 ? 1 : -1);
    }
};

} // namespace nachia

namespace nachia{

template<class Modint>
Modint PolynomialInterpolationOnePoint(
    std::vector<Modint> f,
    Modint x
) {
    int n = f.size();
    auto comb = nachia::Comb<Modint>(n);
    Modint q = 1;
    for(int i=n-1; i>=0; i--){
        f[i] *= q * comb.invFactorial(n-1-i);
        if((n-1-i)%2 == 1) f[i] = -f[i];
        q *= x-Modint::raw(i);
    }
    q = 1;
    Modint ans = 0;
    for(int i=0; i<n; i++){
        ans += f[i] * q * comb.invFactorial(i);
        q *= x-Modint::raw(i);
    }
    return ans;
}

} // namespace nachia

namespace nachia{

// r.val() != 1
template<class Modint>
Modint GeometricPolynomialPrefixSumLimit(
    std::vector<Modint> f,
    Modint r
){
    if(f.size() == 0) return Modint(0);
    int d = f.size() - 1;
    auto comb = nachia::Comb<Modint>(d+1);
    if(r.val() == 0) return f[0];
    Modint rp = 1;
    std::vector<Modint> q(d+2);
    for(int k=0; k<=d; k++){ q[k+1] = q[k] + (f[k] *= rp); rp *= r; }
    if(r.val() == 1) return Modint(0);
    Modint c = 0;
    rp = 1;
    int Z = d + 10;
    for(int k=0; k<=d; k++){ c += comb(d+1,k) * rp * q[d-k+1]; rp *= -r; }
    return c * (Modint(1)-r).inv().pow(d+1);
}

template<class Modint>
Modint GeometricPolynomialPrefixSum(
    std::vector<Modint> f,
    Modint r,
    unsigned long long n
){
    if(f.size() == 0 || n == 0) return Modint(0);
    int d = f.size() - 1;
    auto comb = nachia::Comb<Modint>(d+1);
    if(r.val() == 0) return f[0];
    Modint rp = 1;
    std::vector<Modint> q(d+2);
    for(int k=0; k<=d; k++){ q[k+1] = q[k] + f[k] * rp; rp *= r; }
    if(r.val() == 1) return PolynomialInterpolationOnePoint(std::move(q), Modint(n));
    Modint c = 0;
    rp = 1;
    for(int k=0; k<=d; k++){ c += comb(d+1,k) * rp * q[d-k+1]; rp *= -r; }
    c *= (Modint(1)-r).inv().pow(d+1);
    Modint rinv = r.inv();
    rp = 1;
    for(int i=0; i<=d; i++){ f[i] = (q[i+1] - c) * rp; rp *= rinv; }
    return PolynomialInterpolationOnePoint(std::move(f), Modint(n-1)) * r.pow(n-1) + c;
}

} // namespace nachia

Modint f(i64 K, i64 N){
    Modint n = N;
    Modint k = K;

    if(K == 1){
        return n + n * (n-1) / 2 + n * (n-1) * (n-2) / 6;
    }
    
    vector<Modint> d(K+1);
    rep(i,K+1) d[i] = Modint(i).pow(K);
    vector<Modint> e(3);
    rep(i,3) e[i] = k.inv() * i + 1;
    
    Modint c1 = nachia::GeometricPolynomialPrefixSum<Modint>(d, 1, N);
    Modint c2 = nachia::GeometricPolynomialPrefixSum<Modint>(d, k.inv(), N);
    Modint c3 = nachia::GeometricPolynomialPrefixSum<Modint>(e, k, N);
    return c3 + (c2 * k.pow(N-1) - c1) / (k - 1);
}

void testcase(){
    i64 K, L, R; cin >> K >> L >> R;
    auto ans = f(K, R+1) - f(K, L);
    cout << ans.val() << "\n";
}

int main(){
    ios::sync_with_stdio(false); cin.tie(nullptr);
    testcase();
    return 0;
}
0