結果

問題 No.3080 Colonies on Line
ユーザー 👑 SPD_9X2
提出日時 2025-03-29 02:47:02
言語 C++23
(gcc 13.3.0 + boost 1.87.0)
結果
AC  
実行時間 2,029 ms / 6,500 ms
コード長 16,220 bytes
コンパイル時間 4,915 ms
コンパイル使用メモリ 317,232 KB
実行使用メモリ 16,556 KB
最終ジャッジ日時 2025-03-29 02:47:33
合計ジャッジ時間 30,933 ms
ジャッジサーバーID
（参考情報） 		judge5 / judge1
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ファイルパターン 結果
sample AC * 3
other AC * 35
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ソースコード

diff #
プレゼンテーションモードにする

#line 2 "modint.hpp"
#include <cassert>
#include <iostream>
#include <set>
#include <vector>
template <int md> struct ModInt {
    using lint = long long;
    constexpr static int mod() { return md; }
    static int get_primitive_root() {
        static int primitive_root = 0;
        if (!primitive_root) {
            primitive_root = [&]() {
                std::set<int> fac;
                int v = md - 1;
                for (lint i = 2; i * i <= v; i++)
                    while (v % i == 0) fac.insert(i), v /= i;
                if (v > 1) fac.insert(v);
                for (int g = 1; g < md; g++) {
                    bool ok = true;
                    for (auto i : fac)
                        if (ModInt(g).pow((md - 1) / i) == 1) {
                            ok = false;
                            break;
                        }
                    if (ok) return g;
                }
                return -1;
            }();
        }
        return primitive_root;
    }
    int val_;
    int val() const noexcept { return val_; }
    constexpr ModInt() : val_(0) {}
    constexpr ModInt &_setval(lint v) { return val_ = (v >= md ? v - md : v), *this; }
    constexpr ModInt(lint v) { _setval(v % md + md); }
    constexpr explicit operator bool() const { return val_ != 0; }
    constexpr ModInt operator+(const ModInt &x) const {
        return ModInt()._setval((lint)val_ + x.val_);
    }
    constexpr ModInt operator-(const ModInt &x) const {
        return ModInt()._setval((lint)val_ - x.val_ + md);
    }
    constexpr ModInt operator*(const ModInt &x) const {
        return ModInt()._setval((lint)val_ * x.val_ % md);
    }
    constexpr ModInt operator/(const ModInt &x) const {
        return ModInt()._setval((lint)val_ * x.inv().val() % md);
    }
    constexpr ModInt operator-() const { return ModInt()._setval(md - val_); }
    constexpr ModInt &operator+=(const ModInt &x) { return *this = *this + x; }
    constexpr ModInt &operator-=(const ModInt &x) { return *this = *this - x; }
    constexpr ModInt &operator*=(const ModInt &x) { return *this = *this * x; }
    constexpr ModInt &operator/=(const ModInt &x) { return *this = *this / x; }
    friend constexpr ModInt operator+(lint a, const ModInt &x) { return ModInt(a) + x; }
    friend constexpr ModInt operator-(lint a, const ModInt &x) { return ModInt(a) - x; }
    friend constexpr ModInt operator*(lint a, const ModInt &x) { return ModInt(a) * x; }
    friend constexpr ModInt operator/(lint a, const ModInt &x) { return ModInt(a) / x; }
    constexpr bool operator==(const ModInt &x) const { return val_ == x.val_; }
    constexpr bool operator!=(const ModInt &x) const { return val_ != x.val_; }
    constexpr bool operator<(const ModInt &x) const {
        return val_ < x.val_;
    } // To use std::map<ModInt, T>
    friend std::istream &operator>>(std::istream &is, ModInt &x) {
        lint t;
        return is >> t, x = ModInt(t), is;
    }
    constexpr friend std::ostream &operator<<(std::ostream &os, const ModInt &x) {
        return os << x.val_;
    }
    constexpr ModInt pow(lint n) const {
        ModInt ans = 1, tmp = *this;
        while (n) {
            if (n & 1) ans *= tmp;
            tmp *= tmp, n >>= 1;
        }
        return ans;
    }
    static constexpr int cache_limit = std::min(md, 1 << 21);
    static std::vector<ModInt> facs, facinvs, invs;
    constexpr static void _precalculation(int N) {
        const int l0 = facs.size();
        if (N > md) N = md;
        if (N <= l0) return;
        facs.resize(N), facinvs.resize(N), invs.resize(N);
        for (int i = l0; i < N; i++) facs[i] = facs[i - 1] * i;
        facinvs[N - 1] = facs.back().pow(md - 2);
        for (int i = N - 2; i >= l0; i--) facinvs[i] = facinvs[i + 1] * (i + 1);
        for (int i = N - 1; i >= l0; i--) invs[i] = facinvs[i] * facs[i - 1];
    }
    constexpr ModInt inv() const {
        if (this->val_ < cache_limit) {
            if (facs.empty()) facs = {1}, facinvs = {1}, invs = {0};
            while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2);
            return invs[this->val_];
        } else {
            return this->pow(md - 2);
        }
    }
    constexpr ModInt fac() const {
        while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2);
        return facs[this->val_];
    }
    constexpr ModInt facinv() const {
        while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2);
        return facinvs[this->val_];
    }
    constexpr ModInt doublefac() const {
        lint k = (this->val_ + 1) / 2;
        return (this->val_ & 1) ? ModInt(k * 2).fac() / (ModInt(2).pow(k) * ModInt(k).fac())
                                : ModInt(k).fac() * ModInt(2).pow(k);
    }
    constexpr ModInt nCr(int r) const {
        if (r < 0 or this->val_ < r) return ModInt(0);
        return this->fac() * (*this - r).facinv() * ModInt(r).facinv();
    }
    constexpr ModInt nPr(int r) const {
        if (r < 0 or this->val_ < r) return ModInt(0);
        return this->fac() * (*this - r).facinv();
    }
    static ModInt binom(int n, int r) {
        static long long bruteforce_times = 0;
        if (r < 0 or n < r) return ModInt(0);
        if (n <= bruteforce_times or n < (int)facs.size()) return ModInt(n).nCr(r);
        r = std::min(r, n - r);
        ModInt ret = ModInt(r).facinv();
        for (int i = 0; i < r; ++i) ret *= n - i;
        bruteforce_times += r;
        return ret;
    }
    // Multinomial coefficient, (k_1 + k_2 + ... + k_m)! / (k_1! k_2! ... k_m!)
    // Complexity: O(sum(ks))
    template <class Vec> static ModInt multinomial(const Vec &ks) {
        ModInt ret{1};
        int sum = 0;
        for (int k : ks) {
            assert(k >= 0);
            ret *= ModInt(k).facinv(), sum += k;
        }
        return ret * ModInt(sum).fac();
    }
    // Catalan number, C_n = binom(2n, n) / (n + 1)
    // C_0 = 1, C_1 = 1, C_2 = 2, C_3 = 5, C_4 = 14, ...
    // https://oeis.org/A000108
    // Complexity: O(n)
    static ModInt catalan(int n) {
        if (n < 0) return ModInt(0);
        return ModInt(n * 2).fac() * ModInt(n + 1).facinv() * ModInt(n).facinv();
    }
    ModInt sqrt() const {
        if (val_ == 0) return 0;
        if (md == 2) return val_;
        if (pow((md - 1) / 2) != 1) return 0;
        ModInt b = 1;
        while (b.pow((md - 1) / 2) == 1) b += 1;
        int e = 0, m = md - 1;
        while (m % 2 == 0) m >>= 1, e++;
        ModInt x = pow((m - 1) / 2), y = (*this) * x * x;
        x *= (*this);
        ModInt z = b.pow(m);
        while (y != 1) {
            int j = 0;
            ModInt t = y;
            while (t != 1) j++, t *= t;
            z = z.pow(1LL << (e - j - 1));
            x *= z, z *= z, y *= z;
            e = j;
        }
        return ModInt(std::min(x.val_, md - x.val_));
    }
};
template <int md> std::vector<ModInt<md>> ModInt<md>::facs = {1};
template <int md> std::vector<ModInt<md>> ModInt<md>::facinvs = {1};
template <int md> std::vector<ModInt<md>> ModInt<md>::invs = {0};
using ModInt998244353 = ModInt<998244353>;
// using mint = ModInt<998244353>;
// using mint = ModInt<1000000007>;
#line 3 "convolution/ntt.hpp"
#include <algorithm>
#include <array>
#line 7 "convolution/ntt.hpp"
#include <tuple>
#line 9 "convolution/ntt.hpp"
// CUT begin
// Integer convolution for arbitrary mod
// with NTT (and Garner's algorithm) for ModInt / ModIntRuntime class.
// We skip Garner's algorithm if `skip_garner` is true or mod is in `nttprimes`.
// input: a (size: n), b (size: m)
// return: vector (size: n + m - 1)
template <typename MODINT>
std::vector<MODINT> nttconv(std::vector<MODINT> a, std::vector<MODINT> b, bool skip_garner);
constexpr int nttprimes[3] = {998244353, 167772161, 469762049};
// Integer FFT (Fast Fourier Transform) for ModInt class
// (Also known as Number Theoretic Transform, NTT)
// is_inverse: inverse transform
// ** Input size must be 2^n **
template <typename MODINT> void ntt(std::vector<MODINT> &a, bool is_inverse = false) {
    int n = a.size();
    if (n == 1) return;
    static const int mod = MODINT::mod();
    static const MODINT root = MODINT::get_primitive_root();
    assert(__builtin_popcount(n) == 1 and (mod - 1) % n == 0);
    static std::vector<MODINT> w{1}, iw{1};
    for (int m = w.size(); m < n / 2; m *= 2) {
        MODINT dw = root.pow((mod - 1) / (4 * m)), dwinv = 1 / dw;
        w.resize(m * 2), iw.resize(m * 2);
        for (int i = 0; i < m; i++) w[m + i] = w[i] * dw, iw[m + i] = iw[i] * dwinv;
    }
    if (!is_inverse) {
        for (int m = n; m >>= 1;) {
            for (int s = 0, k = 0; s < n; s += 2 * m, k++) {
                for (int i = s; i < s + m; i++) {
                    MODINT x = a[i], y = a[i + m] * w[k];
                    a[i] = x + y, a[i + m] = x - y;
                }
            }
        }
    } else {
        for (int m = 1; m < n; m *= 2) {
            for (int s = 0, k = 0; s < n; s += 2 * m, k++) {
                for (int i = s; i < s + m; i++) {
                    MODINT x = a[i], y = a[i + m];
                    a[i] = x + y, a[i + m] = (x - y) * iw[k];
                }
            }
        }
        int n_inv = MODINT(n).inv().val();
        for (auto &v : a) v *= n_inv;
    }
}
template <int MOD>
std::vector<ModInt<MOD>> nttconv_(const std::vector<int> &a, const std::vector<int> &b) {
    int sz = a.size();
    assert(a.size() == b.size() and __builtin_popcount(sz) == 1);
    std::vector<ModInt<MOD>> ap(sz), bp(sz);
    for (int i = 0; i < sz; i++) ap[i] = a[i], bp[i] = b[i];
    ntt(ap, false);
    if (a == b)
        bp = ap;
    else
        ntt(bp, false);
    for (int i = 0; i < sz; i++) ap[i] *= bp[i];
    ntt(ap, true);
    return ap;
}
long long garner_ntt_(int r0, int r1, int r2, int mod) {
    using mint2 = ModInt<nttprimes[2]>;
    static const long long m01 = 1LL * nttprimes[0] * nttprimes[1];
    static const long long m0_inv_m1 = ModInt<nttprimes[1]>(nttprimes[0]).inv().val();
    static const long long m01_inv_m2 = mint2(m01).inv().val();
    int v1 = (m0_inv_m1 * (r1 + nttprimes[1] - r0)) % nttprimes[1];
    auto v2 = (mint2(r2) - r0 - mint2(nttprimes[0]) * v1) * m01_inv_m2;
    return (r0 + 1LL * nttprimes[0] * v1 + m01 % mod * v2.val()) % mod;
}
template <typename MODINT>
std::vector<MODINT> nttconv(std::vector<MODINT> a, std::vector<MODINT> b, bool skip_garner) {
    if (a.empty() or b.empty()) return {};
    int sz = 1, n = a.size(), m = b.size();
    while (sz < n + m) sz <<= 1;
    if (sz <= 16) {
        std::vector<MODINT> ret(n + m - 1);
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) ret[i + j] += a[i] * b[j];
        }
        return ret;
    }
    int mod = MODINT::mod();
    if (skip_garner or
        std::find(std::begin(nttprimes), std::end(nttprimes), mod) != std::end(nttprimes)) {
        a.resize(sz), b.resize(sz);
        if (a == b) {
            ntt(a, false);
            b = a;
        } else {
            ntt(a, false), ntt(b, false);
        }
        for (int i = 0; i < sz; i++) a[i] *= b[i];
        ntt(a, true);
        a.resize(n + m - 1);
    } else {
        std::vector<int> ai(sz), bi(sz);
        for (int i = 0; i < n; i++) ai[i] = a[i].val();
        for (int i = 0; i < m; i++) bi[i] = b[i].val();
        auto ntt0 = nttconv_<nttprimes[0]>(ai, bi);
        auto ntt1 = nttconv_<nttprimes[1]>(ai, bi);
        auto ntt2 = nttconv_<nttprimes[2]>(ai, bi);
        a.resize(n + m - 1);
        for (int i = 0; i < n + m - 1; i++)
            a[i] = garner_ntt_(ntt0[i].val(), ntt1[i].val(), ntt2[i].val(), mod);
    }
    return a;
}
template <typename MODINT>
std::vector<MODINT> nttconv(const std::vector<MODINT> &a, const std::vector<MODINT> &b) {
    return nttconv<MODINT>(a, b, false);
}
#line 5 "formal_power_series/coeff_of_rational_function.hpp"
// CUT begin
// Calculate [x^N](num(x) / den(x))
// - Coplexity: O(LlgLlgN) ( L = size(num) + size(den) )
// - Reference: `Bostan–Mori algorithm` <https://qiita.com/ryuhe1/items/da5acbcce4ac1911f47a>
template <typename Tp>
Tp coefficient_of_rational_function(long long N, std::vector<Tp> num, std::vector<Tp> den) {
    assert(N >= 0);
    while (den.size() and den.back() == 0) den.pop_back();
    assert(den.size());
    int h = 0;
    while (den[h] == 0) h++;
    N += h;
    den.erase(den.begin(), den.begin() + h);
    if (den.size() == 1) return N < int(num.size()) ? num[N] / den[0] : 0;
    while (N) {
        std::vector<Tp> g = den;
        for (size_t i = 1; i < g.size(); i += 2) { g[i] = -g[i]; }
        auto conv_num_g = nttconv(num, g);
        num.resize((conv_num_g.size() + 1 - (N & 1)) / 2);
        for (size_t i = 0; i < num.size(); i++) { num[i] = conv_num_g[i * 2 + (N & 1)]; }
        auto conv_den_g = nttconv(den, g);
        for (size_t i = 0; i < den.size(); i++) { den[i] = conv_den_g[i * 2]; }
        N >>= 1;
    }
    return num[0] / den[0];
}
// Find the n-th term of the sequence (0-ORIGIN)
// Complexity: O(K lg K \log N)
// ainit = [a_0, a_1,..., ]
// c[0] = 1, \sum_j a_{i - j} * c_j = 0
template <typename Tp>
Tp find_kth_term(std::vector<Tp> ainit, const std::vector<Tp> c, long long n) {
    assert(ainit.size() + 1 == c.size());
    auto a = nttconv(ainit, c);
    a.resize(ainit.size());
    return coefficient_of_rational_function(n, a, c);
}
#line 3 "formal_power_series/test/kth_term_of_linearly_recurrent_sequence.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/kth_term_of_linearly_recurrent_sequence"
#line 5 "formal_power_series/test/kth_term_of_linearly_recurrent_sequence.test.cpp"
using namespace std;
#include <bits/stdc++.h>
#include <iostream>
#include <limits>
#include <numeric>
#include <type_traits>
#include <bitset>
#include <map>
#include <unordered_map>
#include <set>
#include <random>
using namespace std;
using ll = long long;
#define rep(i,n,m) for(ll (i)=(n);(i)<(m);(i)++)
#define rrep(i,n,m) for(ll (i)=(n);(i)>(m);(i)--)
#define LAMBDA2(x, y, ...) ([&](auto&& x, auto&& y) -> decltype(auto) { return __VA_ARGS__; })
#define DivFloor(...) LAMBDA2(x, y, x / y - ((x ^ y) < 0 && x % y != 0))(__VA_ARGS__)
#define DivCeil(...) LAMBDA2(x, y, x / y + ((x ^ y) >= 0 && x % y != 0))(__VA_ARGS__)
const ll inf = 1e9;
const ll INF = 1e18;
using pll = pair<ll,ll>;
using vll = vector<ll>;
using vvll = vector<vector<ll>>;
using vpll = vector<pair<ll,ll>>;
using sll = set<ll>;
using mll = map<ll,ll>;
void pline(vector<int> lis){
    rep(i,0,lis.size()){
        printf ("%d",lis[i]);
        if (i != lis.size()-1) printf(" ");
        else printf("\n");
    }
}
void pline(vector<ll> lis){
    rep(i,0,lis.size()){
        printf ("%lld",lis[i]);
        if (i != lis.size()-1) printf(" ");
        else printf("\n");
    }
}
void pline(vector<bool> lis){
    rep(i,0,lis.size()){
        if (lis[i]) printf("t");
        else printf("f");
        if (i != lis.size()-1) printf(" ");
        else printf("\n");
    }
}
void pline(set<ll> lis){
    for (ll x : lis){
        cout << x << " ";
    }
    cout << endl;
}
void pline(vector<pair<ll,ll>> lis){
    rep(i,0,lis.size()){
        printf ("/%lld,%lld/",lis[i].first,lis[i].second);
        if (i != lis.size()-1) printf(" ");
        else printf("\n");
    }
}
int main() {
    ll N,k;
    cin >> N >> k;
    vector<ModInt<998244353>> A(k+3,0), B(2*k+4,0);
    A[k+2] -= 1;
    A[2] = 1;
    
    B[0] += 1;
    B[1] -= 4;
    B[2] += 5;
    B[3] -= 2;
    B[k+1] += 1;
    B[k+2] -= 3;
    B[k+3] += 2;
    B[2*k+2] += 1;
    B[2*k+3] -= 1;
    
    auto nth_coeff = coefficient_of_rational_function(N, A, B);
    cout << nth_coeff+1 << endl;
}
/*
https://yukicoder.me/problems/no/3080
p = ( (x-x^(k+1))(p+(( (x^(k+1)*p+1) ) / (1-x))) ) / (1-x)
q = (( (x^(k+1)*p+1) ) / (1-x))
https://www.wolframalpha.com/input?i=Solve+for+p%3A+p+%3D+%28+%28x-x%5E%28k%2B1%29%29%28p%2B%28%28+%28x%5E%28k%2B1%29*p%2B1%29+%29+%2F+%281
    -x%29%29%29+%29+%2F+%281-x%29&lang=ja
答は 1 + [x^(N-1)] P(x) / (1-x)
https://www.wolframalpha.com/input?i=%28%281+-+3+x+%2B+2+x%5E2+%2B+x%5E%281+%2B+k%29+-+2+x%5E%282+%2B+k%29+%2B+x%5E%282+%2B+2+k%29%29%281
    -x%29%29&lang=ja
*/
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