結果

問題 No.2883 K-powered Sum of Fibonacci
ユーザー kusaf_kusaf_
提出日時 2024-09-11 00:02:04
言語 C++23
(gcc 12.3.0 + boost 1.83.0)
結果
AC  
実行時間 4 ms / 3,000 ms
コード長 14,292 bytes
コンパイル時間 5,116 ms
コンパイル使用メモリ 300,156 KB
実行使用メモリ 6,944 KB
最終ジャッジ日時 2024-09-11 00:02:11
合計ジャッジ時間 6,595 ms
ジャッジサーバーID
（参考情報） 		judge1 / judge3
このコードへのチャレンジ
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テストケース

テストケース表示
入力 結果 実行時間
実行使用メモリ
testcase_00 AC 2 ms
6,812 KB
testcase_01 AC 2 ms
6,944 KB
testcase_02 AC 2 ms
6,944 KB
testcase_03 AC 2 ms
6,940 KB
testcase_04 AC 2 ms
6,940 KB
testcase_05 AC 3 ms
6,940 KB
testcase_06 AC 2 ms
6,940 KB
testcase_07 AC 2 ms
6,944 KB
testcase_08 AC 3 ms
6,944 KB
testcase_09 AC 2 ms
6,940 KB
testcase_10 AC 2 ms
6,940 KB
testcase_11 AC 3 ms
6,944 KB
testcase_12 AC 3 ms
6,940 KB
testcase_13 AC 2 ms
6,944 KB
testcase_14 AC 3 ms
6,944 KB
testcase_15 AC 3 ms
6,944 KB
testcase_16 AC 2 ms
6,944 KB
testcase_17 AC 3 ms
6,944 KB
testcase_18 AC 2 ms
6,944 KB
testcase_19 AC 3 ms
6,940 KB
testcase_20 AC 3 ms
6,940 KB
testcase_21 AC 3 ms
6,944 KB
testcase_22 AC 3 ms
6,944 KB
testcase_23 AC 4 ms
6,940 KB
testcase_24 AC 4 ms
6,940 KB
testcase_25 AC 3 ms
6,940 KB
testcase_26 AC 4 ms
6,944 KB
testcase_27 AC 4 ms
6,944 KB
testcase_28 AC 4 ms
6,944 KB
testcase_29 AC 3 ms
6,944 KB
testcase_30 AC 2 ms
6,940 KB
testcase_31 AC 2 ms
6,940 KB
testcase_32 AC 2 ms
6,940 KB
testcase_33 AC 2 ms
6,940 KB
testcase_34 AC 2 ms
6,944 KB
testcase_35 AC 2 ms
6,944 KB
testcase_36 AC 2 ms
6,944 KB
testcase_37 AC 2 ms
6,944 KB
testcase_38 AC 2 ms
6,940 KB
testcase_39 AC 2 ms
6,944 KB
testcase_40 AC 2 ms
6,940 KB
testcase_41 AC 2 ms
6,940 KB
testcase_42 AC 3 ms
6,944 KB
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 # 

#include <bits/stdc++.h>
#include <atcoder/convolution>
using namespace std;
using namespace atcoder;
using ll = long long;
using mint = modint998244353;

#define FAST
template<ll MOD = 998244353, typename T = mint> struct FPS : vector<T> {
  using vector<T>::vector;
  using vector<T>::operator=;
  FPS pre(int deg) const {
    FPS r(begin(*this), begin(*this) + min((int)this->size(), deg));
    if((int)r.size() < deg) { r.resize(deg); }
    return r;
  }
  FPS rev(int deg = -1) const {
    FPS r(*this);
    if(deg != -1) { r.resize(deg, T(0)); }
    ranges::reverse(r);
    return r;
  }
  void shrink() {
    while(this->size() && this->back() == T(0)) { this->pop_back(); }
  }
  FPS operator+(const FPS &f) const { return FPS(*this) += f; }
  FPS operator+(const T &x) const { return FPS(*this) += x; }
  FPS operator-(const FPS &f) const { return FPS(*this) -= f; }
  FPS operator-(const T &x) const { return FPS(*this) -= x; }
  FPS operator*(const FPS &f) const { return FPS(*this) *= f; }
  template<typename I> FPS operator*(const vector<pair<I, T>> &f) const { return FPS(*this) *= f; }
  template<typename I> FPS operator*(const pair<I, T> &f) const { return FPS(*this) *= f; }
  FPS operator*(const T &x) const { return FPS(*this) *= x; }
  FPS operator/(const FPS &f) const { return FPS(*this) /= f; }
  template<typename I> FPS operator/(vector<pair<I, T>> &f) const { return FPS(*this) /= f; }
  template<typename I> FPS operator/(const pair<I, T> &f) const { return FPS(*this) /= f; }
  FPS operator/(const T &x) const { return FPS(*this) /= x; }
  FPS operator%(const FPS &f) const { return FPS(*this) %= f; }
  FPS &operator+=(const FPS &f) {
    if(f.size() > this->size()) { this->resize(f.size()); }
    for(int i = 0; i < (int)f.size(); i++) { (*this)[i] += f[i]; }
    return *this;
  }
  FPS &operator-=(const FPS &f) {
    if(f.size() > this->size()) { this->resize(f.size()); }
    for(int i = 0; i < (int)f.size(); i++) { (*this)[i] -= f[i]; }
    return *this;
  }
  FPS &operator*=(const FPS &f) {
#ifdef FAST
    *this = convolution(*this, f);
#else
    const int n = this->size(), m = f.size();
    vector<ll> a(n), b(m);
    static constexpr ll MOD1 = 754974721, MOD2 = 167772161, MOD3 = 469762049;
    static constexpr ll M1_M2 = internal::inv_gcd(MOD1, MOD2).second, M12_M3 = internal::inv_gcd(MOD1 * MOD2, MOD3).second, M12 = (MOD1 * MOD2) % MOD;
    for(int i = 0; i < n; i++) { a[i] = (*this)[i].val(); }
    for(int i = 0; i < m; i++) { b[i] = f[i].val(); }
    vector<ll> x = convolution<MOD1>(a, b), y = convolution<MOD2>(a, b), z = convolution<MOD3>(a, b);
    vector<T> c(n + m - 1);
    for(int i = 0; i < n + m - 1; i++) {
      ll v1 = (y[i] - x[i]) * M1_M2 % MOD2;
      if(v1 < 0) { v1 += MOD2; }
      ll v2 = (z[i] - (x[i] + MOD1 * v1) % MOD3) * M12_M3 % MOD3;
      if(v2 < 0) { v2 += MOD3; }
      c[i] = x[i] + MOD1 * v1 + M12 * v2;
    }
    *this = c;
#endif
    return *this;
  }
  template<typename I> FPS &operator*=(const vector<pair<I, T>> &f) {
    const int n = this->size() - 1, m = f.back().first;
    FPS r(n + m + 1, 0);
    for(int i = 0; i <= n; i++) {
      for(auto &[j, c] : f) { r[i + j] += (*this)[i] * c; }
    }
    return *this = r;
  }
  template<typename I> FPS &operator*=(const pair<I, T> &f) {  // *(cx^d + 1)
    const int n = this->size();
    auto [d, c] = f;
    for(int i = n - d - 1; i >= 0; i--) { (*this)[i + d] += (*this)[i] * c; }
    return *this;
  }
  FPS &operator/=(const FPS &f) {
    if(this->size() < f.size()) {
      this->clear();
      return *this;
    }
    return *this *= f.inv();
  }
  template<typename I> FPS &operator/=(vector<pair<I, T>> &f) {
    ranges::sort(f, [&](auto x, auto y) { return x.first > y.first; });
    const ll n = this->size() - 1, m = f[0].first;
    FPS r(n - m + 1, 0);
    for(int i = n - m; i >= 0; i--) {
      r[i] = (*this)[i + m] / f[0].second;
      for(auto &[j, c] : f) { (*this)[i + j] -= r[i] * c; }
    }
    return *this = r;
  }
  template<typename I> FPS &operator/=(const pair<I, T> &f) {  // /(cx^d + 1)
    const int n = this->size();
    auto [d, c] = f;
    for(int i = 0; i < n - d; i++) { (*this)[i + d] -= (*this)[i] * c; }
    return *this;
  }
  FPS &operator%=(const FPS &f) { return *this -= *this / f * f; }
  pair<FPS, FPS> div_mod(const FPS &f) {
    FPS g = *this / f;
    return {g, *this - g * f};
  }
  FPS operator-() {
    FPS r(this->size());
    for(int i = 0; i < (this->size()); i++) { r[i] = -(*this)[i]; }
    return r;
  }
  FPS &operator+=(const T &x) {
    if(this->empty()) { this->resize(1); }
    (*this)[0] += x;
    return *this;
  }
  FPS &operator-=(const T &x) {
    if(this->empty()) { this->resize(1); }
    (*this)[0] -= x;
    return *this;
  }
  FPS &operator*=(const T &x) {
    for(int i = 0; i < (int)this->size(); i++) { (*this)[i] *= x; }
    return *this;
  }
  FPS &operator/=(const T &x) {
    for(int i = 0; i < (int)this->size(); i++) { (*this)[i] /= x; }
    return *this;
  }
  FPS operator>>(ll sz) {
    if((int)this->size() <= sz) { return {}; }
    FPS r(*this);
    r.erase(r.begin(), r.begin() + sz);
    return r;
  }
  FPS operator<<(ll sz) {
    FPS r(*this);
    r.insert(r.begin(), sz, T(0));
    return r;
  }
  FPS dot(const FPS &f) const {
    FPS r(min(this->size(), f.size()));
    for(int i = 0; i < r.size(); i++) { r[i] = (*this)[i] * f[i]; }
    return r;
  }
  T operator()(T x) const {
    T r = 0, w = 1;
    for(auto &i : (*this)) {
      r += w * i;
      w *= x;
    }
    return r;
  }
  FPS diff() const {
    const int n = this->size();
    FPS r(n);
    for(int i = 1; i < n; i++) { r[i - 1] = (*this)[i] * T(i); }
    r[n - 1] = 0;
    return r;
  }
  FPS integral() const {
    const int n = this->size();
    vector<T> inv(n);
    inv[1] = 1;
    for(int i = 2; i < n; i++) { inv[i] = -inv[MOD % i] * (MOD / i); }
    FPS r(n);
    for(int i = n - 2; i >= 0; i--) { r[i + 1] = (*this)[i] * inv[i + 1]; }
    r[0] = 0;
    return r;
  }
  FPS inv(ll deg = -1) const {
    const int n = this->size();
    if(deg == -1) { deg = n; }
    assert(n && (*this)[0] != T(0));
    FPS res{(*this)[0].inv()};
#ifdef FAST
    while((int)res.size() < deg) {
      int d = res.size();
      FPS f(this->begin(), this->begin() + min(n, d * 2)), g(res);
      f.resize(d * 2);
      g.resize(d * 2);
      internal::butterfly(f);
      internal::butterfly(g);
      for(int i = 0; i < d * 2; i++) { f[i] *= g[i]; }
      internal::butterfly_inv(f);
      f.erase(f.begin(), f.begin() + d);
      f.resize(d * 2);
      internal::butterfly(f);
      for(int i = 0; i < d * 2; i++) { f[i] *= g[i]; }
      internal::butterfly_inv(f);
      T iz = T(d * 2).inv();
      iz *= -iz;
      for(int i = 0; i < d; i++) { f[i] *= iz; }
      res.insert(res.end(), f.begin(), f.begin() + d);
    }
#else
    for(int i = 1; i < deg; i <<= 1) { res = (res + res - res * res * pre(i << 1)).pre(i << 1); }
#endif
    return res.pre(deg);
  }
  FPS log(ll deg = -1) const {
    assert((*this)[0] == T(1));
    if(deg == -1) { deg = this->size(); }
    return (this->diff() * this->inv(deg)).pre(deg).integral();
  }
  FPS sqrt(ll deg = -1) {
    const int n = this->size();
    if(deg == -1) { deg = n; }
    if((*this)[0] == T(0)) {
      for(int i = 1; i < n; i++) {
        if((*this)[i] != T(0)) {
          if(i & 1) { return {}; }
          if(deg - i / 2 <= 0) { break; }
          auto r = (*this >> i).sqrt(deg - i / 2);
          if(r.empty()) { return {}; }
          r = r << (i / 2);
          if((int)r.size() < deg) { r.resize(deg, T(0)); }
          return r;
        }
      }
      return FPS(deg, 0);
    }
    auto mod_sqrt = [&](const ll &a) -> ll {
      ll m = MOD - 1, e = 0;
      if(!a) { return 0; }
      if(MOD == 2) { return a; }
      if(T(a).pow(m >> 1) != 1) { return -1; }
      T b = 1;
      while(b.pow(m >> 1) == 1) { b++; }
      while(~m & 1) {
        m >>= 1;
        e++;
      }
      T x = T(a).pow((m - 1) >> 1), y = T(a) * x * x, z = T(b).pow(m);
      x *= a;
      while(y != 1) {
        ll j = 0;
        T t = y;
        while(t != 1) {
          j++;
          t *= t;
        }
        z = z.pow(1LL << (e - j - 1));
        x *= z;
        z *= z;
        y *= z;
        e = j;
      }
      return x.val();
    };
    auto sq = T(mod_sqrt((*this)[0].val()));
    if(sq * sq != (*this)[0]) { return {}; }
    FPS r{sq};
    T inv2 = T(1) / T(2);
    for(int i = 1; i < deg; i <<= 1) { r = (r + pre(i << 1) * r.inv(i << 1)) * inv2; }
    return r.pre(deg);
  }
  FPS sqrt(const function<T(T)> &get_sqrt, ll deg = -1) { return sqrt(deg, get_sqrt); }
  FPS exp(ll deg = -1) const {
    const int n = this->size();
    assert((*this)[0] == T(0));
    if(deg == -1) { deg = n; }
#ifdef FAST
    FPS inv;
    inv.reserve(deg);
    inv.emplace_back(T(0));
    inv.emplace_back(T(1));
    auto internal_integral = [&](FPS &f) {
      const int n = f.size();
      while((int)inv.size() <= n) {
        int i = inv.size();
        inv.emplace_back((-inv[MOD % i]) * (MOD / i));
      }
      f.insert(f.begin(), T(0));
      for(int i = 1; i <= n; i++) { f[i] *= inv[i]; }
    };
    auto internal_diff = [](FPS &f) {
      if(f.empty()) { return; }
      f.erase(f.begin());
      T c = 1;
      for(int i = 0; i < (int)f.size(); i++, c++) { f[i] *= c; }
    };
    FPS b{1, 1 < (int)this->size() ? (*this)[1] : 0}, c{1}, z1, z2{1, 1};
    for(int m = 2; m <= deg; m <<= 1) {
      auto y = b;
      y.resize(m * 2);
      internal::butterfly(y);
      z1 = z2;
      FPS z(m);
      for(int i = 0; i < m; i++) { z[i] = y[i] * z1[i]; }
      internal::butterfly_inv(z);
      T si = T(m).inv();
      for(int i = 0; i < m; i++) { z[i] *= si; }
      fill(z.begin(), z.begin() + m / 2, T(0));
      internal::butterfly(z);
      for(int i = 0; i < m; i++) { z[i] *= -z1[i]; }
      internal::butterfly_inv(z);
      for(int i = 0; i < m; i++) { z[i] *= si; }
      c.insert(c.end(), z.begin() + m / 2, z.end());
      z2 = c;
      z2.resize(m * 2);
      internal::butterfly(z2);
      FPS x(this->begin(), this->begin() + min((int)this->size(), m));
      x.resize(m);
      internal_diff(x);
      x.emplace_back(T(0));
      internal::butterfly(x);
      for(int i = 0; i < m; i++) { x[i] *= y[i]; }
      internal::butterfly_inv(x);
      for(int i = 0; i < m; i++) { x[i] *= si; }
      x -= b.diff();
      x.resize(m * 2);
      for(int i = 0; i < m - 1; i++) {
        x[m + i] = x[i];
        x[i] = T(0);
      }
      internal::butterfly(x);
      for(int i = 0; i < m * 2; i++) { x[i] *= z2[i]; }
      internal::butterfly_inv(x);
      T si2 = T(m << 1).inv();
      for(int i = 0; i < m * 2; i++) { x[i] *= si2; }
      x.pop_back();
      internal_integral(x);
      for(int i = m; i < min((int)this->size(), m * 2); i++) { x[i] += (*this)[i]; }
      fill(x.begin(), x.begin() + m, T(0));
      internal::butterfly(x);
      for(int i = 0; i < m * 2; i++) { x[i] *= y[i]; }
      internal::butterfly_inv(x);
      for(int i = 0; i < m * 2; i++) { x[i] *= si2; }
      b.insert(b.end(), x.begin() + m, x.end());
    }
    return b.pre(deg);
#else
    FPS r({T(1)});
    for(int i = 1; i < deg; i <<= 1) { r = (r * (pre(i << 1) + T(1) - r.log(i << 1))).pre(i << 1); }
    return r.pre(deg);
#endif
  }
  FPS pow(ll k) {
    const int n = this->size();
    assert(k >= 0);
    if(k == 0) {
      FPS r(n, T(0));
      r[0] = T(1);
      return r;
    }
    for(int i = 0; i < n; i++) {
      if(i * k > n) { return FPS(n, T(0)); }
      if((*this)[i] != T(0)) {
        T rev = (*this)[i].inv();
        FPS r = (((*this * rev) >> i).log() * k).exp() * ((*this)[i].pow(k));
        r = (r << (i * k)).pre(n);
        if((int)r.size() < n) { r.resize(n, T(0)); }
        return r;
      }
    }
    return *this;
  }
  FPS mod_pow(ll k, FPS f) const {
    FPS modinv = f.rev().inv();
    auto get_div = [&](FPS base) {
      if(base.size() < f.size()) {
        base.clear();
        return base;
      }
      ll n = base.size() - f.size() + 1;
      return (base.rev().pre(n) * modinv.pre(n)).pre(n).rev(n);
    };
    FPS x(*this), r{1};
    while(k > 0) {
      if(k & 1) {
        r *= x;
        r -= get_div(r) * f;
        r.shrink();
      }
      x *= x;
      x -= get_div(x) * f;
      x.shrink();
      k >>= 1;
    }
    return r;
  }
};

mint BostanMori(FPS<> p, FPS<> q, ll n) {
  ll m = max(p.size(), q.size());
  p.resize(m);
  q.resize(m);
  while(n) {
    FPS<> r = q;
    for(ll i = 0; i < ssize(r); i += 2) { r[i] = -r[i]; }
    FPS<> v = q * r, u = p * r;
    for(ll i = n % 2; i < ssize(u); i += 2) { p[i / 2] = u[i]; }
    for(ll i = 0; i < ssize(v); i += 2) { q[i / 2] = v[i]; }
    n /= 2;
  }
  return p[0] / q[0];
}

template<typename T, typename U> mint LinearRecurrence(const vector<T> &ini, const vector<U> &rec, ll n) {
  ll s = ini.size(), k = rec.size();
  assert(s >= k);
  FPS<> p, q(k + 1), a(s);
  q[0] = 1;
  for(ll i = 0; i < k; i++) { q[i + 1] = -rec[i]; }
  for(ll i = 0; i < s; i++) { a[i] = ini[i]; }
  p = (q * a).pre(k);
  return BostanMori(p, q, n);
}

template<typename T> vector<mint> BerlekampMassey(const vector<T> &v) {
  const int N = v.size();
  vector<mint> b, c;
  b.reserve(N + 1);
  c.reserve(N + 1);
  b.push_back(1);
  c.push_back(1);
  mint y = 1;
  for(int ed = 1; ed <= N; ed++) {
    int l = c.size(), m = b.size();
    mint x = 0;
    for(int i = 0; i < l; i++) { x += c[i] * v[ed - l + i]; }
    b.emplace_back(0);
    m++;
    if(x == 0) { continue; }
    mint freq = x / y;
    if(l < m) {
      auto tmp = c;
      c.insert(begin(c), m - l, 0);
      for(int i = 0; i < m; i++) { c[m - i - 1] -= freq * b[m - i - 1]; }
      b = tmp;
      y = x;
    }
    else {
      for(int i = 0; i < m; i++) { c[l - i - 1] -= freq * b[m - i - 1]; }
    }
  }
  c.pop_back();
  for(auto &i : c) { i = -i; }
  ranges::reverse(c);
  return c;
}

template<typename T> mint BMBM(const vector<T> &v, ll n) {
  auto bm = BerlekampMassey(v);
  return LinearRecurrence(v, bm, n);
}

int main() {
  ios::sync_with_stdio(false);
  cin.tie(nullptr);

  ll N, K;
  cin >> N >> K;

  vector<mint> f{1, 1}, v{1, 2};
  for(ll i = 0; i < 1000; i++) {
    f.emplace_back(f[i] + f[i + 1]);
    v.emplace_back(v.back() + f.back().pow(K));
  }

  cout << BMBM(v, --N).val() << "\n";
}
0