結果

問題 No.1559 Next Rational
ユーザー Taiki0715Taiki0715
提出日時 2023-07-18 19:42:23
言語 C++17(clang)
(17.0.6 + boost 1.83.0)
結果
AC  
実行時間 2 ms / 2,000 ms
コード長 12,771 bytes
コンパイル時間 10,224 ms
コンパイル使用メモリ 184,980 KB
実行使用メモリ 5,376 KB
最終ジャッジ日時 2024-09-18 19:02:38
合計ジャッジ時間 8,518 ms
ジャッジサーバーID
（参考情報） 		judge3 / judge2
このコードへのチャレンジ
（要ログイン）

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
5,248 KB
testcase_01 AC 2 ms
5,248 KB
testcase_02 AC 2 ms
5,376 KB
testcase_03 AC 2 ms
5,376 KB
testcase_04 AC 2 ms
5,376 KB
testcase_05 AC 2 ms
5,376 KB
testcase_06 AC 1 ms
5,376 KB
testcase_07 AC 2 ms
5,376 KB
testcase_08 AC 1 ms
5,376 KB
testcase_09 AC 1 ms
5,376 KB
testcase_10 AC 2 ms
5,376 KB
testcase_11 AC 2 ms
5,376 KB
testcase_12 AC 2 ms
5,376 KB
testcase_13 AC 2 ms
5,376 KB
testcase_14 AC 1 ms
5,376 KB
testcase_15 AC 2 ms
5,376 KB
testcase_16 AC 2 ms
5,376 KB
testcase_17 AC 2 ms
5,376 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>
using namespace std;
#if __has_include(<atcoder/all>)
#include <atcoder/all>
using namespace atcoder;
template<int mod>istream &operator>>(istream &is,static_modint<mod> &a){long long b;is>>b;a=b;return is;}
template<int mod>ostream &operator<<(ostream &os,const static_modint<mod> &a){os<<a.val();return os;}
istream &operator>>(istream &is,modint &a){long long b;cin>>b;a=b;return is;}
ostream &operator<<(ostream &os,const modint &a){os<<a.val();return os;}
#endif
using ll=long long;
using ull=unsigned long long;
using P=pair<ll,ll>;
template<typename T>using minque=priority_queue<T,vector<T>,greater<T>>;
template<typename T>bool chmax(T &a,const T &b){return (a<b?(a=b,true):false);}
template<typename T>bool chmin(T &a,const T &b){return (a>b?(a=b,true):false);}
template<typename T1,typename T2>ostream &operator<<(ostream &os,const pair<T1,T2>&p){os<<p.first<<' '<<p.second;return os;}
template<typename T1,typename T2>istream &operator>>(istream &is,pair<T1,T2>&p){is>>p.first>>p.second;return is;}
template<typename T> ostream &operator<<(ostream &os,const vector<T> &a){cout<<'{';for(int i=0;i<a.size();i++){cout<<a[i];if(i+1!=a.size())cout<<',';};cout<<'}';return os;}
template<typename T>istream &operator>>(istream &is,vector<T> &a){for(auto &i:a)is>>i;return is;}
template<typename T1,typename T2>void operator++(pair<T1,T2>&a,int n){a.first++,a.second++;}
template<typename T1,typename T2>void operator--(pair<T1,T2>&a,int n){a.first--,a.second--;}
template<typename T>void operator++(vector<T>&a,int n){for(auto &i:a)i++;}
template<typename T>void operator--(vector<T>&a,int n){for(auto &i:a)i--;}
#define reps(i,a,n) for(int i=(a);i<(n);i++)
#define rep(i,n) reps(i,0,n)
#define all(x) x.begin(),x.end()
#define pcnt(x) __builtin_popcount(x)
ll myceil(ll a,ll b){return (a+b-1)/b;}
template<typename T1,typename ...T2>
void debug(const T1& x,const T2& ...y){
  #ifdef LOCAL
  cout<<x;
  ((cout<<' '<<y),...);
  cout<<'\n';
  #endif
}
void SOLVE();
int main(){
  ios::sync_with_stdio(false);
  cin.tie(nullptr);
  #ifdef LOCAL
  clock_t start=clock();
  #endif
  int testcase=1;
  //cin>>testcase;
  for(int i=0;i<testcase;i++){
    SOLVE();
  }
  #ifdef LOCAL
  cout<<"time:";
  cout<<(clock()-start)/1000;
  cout<<"ms\n";
  #endif
}
constexpr int primitive_root(int x){
  if(x==998244353)return 3;
  return -1;
}
constexpr int countr_zero(unsigned int n){
  int ret=0;
  while(!(n&(1<<ret)))ret++;
  return ret;
}
template<typename T>
struct ntt_root{
  static constexpr int rank2=countr_zero(T::mod()-1);
  int g=primitive_root(T::mod());
  array<T,rank2+1>root,invroot;
  array<T,max(0,rank2-1)>rate2,invrate2;
  array<T,max(0,rank2-2)>rate3,invrate3;
  ntt_root(){
    root[rank2]=T::raw(g).pow((T::mod())>>rank2);
    invroot[rank2]=root[rank2].inv();
    for(int i=rank2-1;i>=0;i--){
      root[i]=root[i+1]*root[i+1];
      invroot[i]=invroot[i+1]*invroot[i+1];
    }
    {
      T prod=1,invprod=1;
      rep(i,rank2-1){
        rate2[i]=root[i+2]*prod;
        invrate2[i]=invroot[i+2]*invprod;
        prod*=invroot[i+2];
        invprod*=root[i+2];
      }
    }
    {
      T prod=1,invprod=1;
      rep(i,rank2-2){
        rate3[i]=root[i+3]*prod;
        invrate3[i]=invroot[i+3]*invprod;
        prod*=invroot[i+3];
        invprod*=root[i+3];
      }
    }
  }
};
template<typename T>
void dft(vector<T>&a){
  int n=a.size();
  static const ntt_root<T> r;
  int h=countr_zero(n);
  ull mod2=(ull)T::mod()*T::mod();
  int len=0;
  while(len<h){
    if(h-len==1){
      T rot=1;
      rep(s,1<<len){
        int of=s*2;
        T u=a[of],v=a[of+1]*rot;
        a[of]=u+v;
        a[of+1]=u-v;
        if(s+1!=1<<len)rot*=r.rate2[__builtin_ctz(~(unsigned int)s)];
      }
      len++;
    }
    else{
      int p=1<<(h-len-2);
      T rot=1,imag=r.root[2];
      rep(s,1<<len){
        T rot2=rot*rot,rot3=rot*rot2;
        int of=s<<(h-len);
        rep(i,p){
          ull a0=a[i+of].val(),a1=(ull)a[i+of+p].val()*rot.val(),a2=(ull)a[i+of+p*2].val()*rot2.val(),a3=(ull)a[i+of+p*3].val()*rot3.val();
          ull m=(ull)T(a1+mod2-a3).val()*imag.val();
          ull k=mod2-a2;
          a[i+of]=a0+a2+a1+a3;
          a[i+of+p]=a0+a2+(mod2*2-a1-a3);
          a[i+of+p*2]=a0+k+m;
          a[i+of+p*3]=a0+k+(mod2-m);
        }
        if(s+1!=1<<len)rot*=r.rate3[__builtin_ctz(~(unsigned int)s)];
      }
      len+=2;
    }
  }
}
template<typename T>
void idft(vector<T>&a){
  int n=a.size();
  static const ntt_root<T> r;
  int h=countr_zero(n);
  ull mod2=(ull)T::mod()*T::mod();
  int len=h;
  while(len){
    if(len==1){
      int p=1<<(h-1);
      T rot=1;
      rep(i,p){
        T u=a[i],v=a[i+p];
        a[i]=u+v;
        a[i+p]=(ull)(u.val()-v.val()+T::mod())*rot.val();
      }
      len--;
    }
    else{
      int p=1<<(h-len);
      T rot=1,imag=r.invroot[2];
      rep(s,1<<(len-2)){
        T rot2=rot*rot,rot3=rot*rot2;
        int of=s<<(h-len+2);
        rep(i,p){
          ull a0=a[i+of].val(),a1=a[i+of+p].val(),a2=a[i+of+p*2].val(),a3=a[i+of+p*3].val();
          ull k=T((T::mod()+a2-a3)*imag.val()).val();
          a[i+of]=a0+a1+a2+a3;
          a[i+of+p]=(a0+T::mod()-a1+k)*rot.val();
          a[i+of+p*2]=(a0+a1+T::mod()*2-a2-a3)*rot2.val();
          a[i+of+p*3]=(a0+T::mod()*2-a1-k)*rot3.val();
        }
        if(s+1!=1<<(len-2))rot*=r.invrate3[countr_zero(~(unsigned int)s)];
      }
      len-=2;
    }
  }
}
template<typename T>
vector<T>ntt_convolution(vector<T> a,vector<T> b){
  int n=a.size(),m=b.size(),s=n+m-1;
  if(min(n,m)<60){
    vector<T>ret(s,0);
    if(n<m)rep(i,m)rep(j,n)ret[i+j]+=a[j]*b[i];
    else rep(i,n)rep(j,m)ret[i+j]+=a[i]*b[j];
    return ret;
  }
  int z=1;
  while(z<=s)z*=2;
  a.resize(z,0);
  b.resize(z,0);
  dft(a),dft(b);
  vector<T>c(z);
  rep(i,z)c[i]=a[i]*b[i];
  idft(c);
  T g=T::raw(z).inv();
  rep(i,s)c[i]*=g;
  return {c.begin(),c.begin()+s};
}
template<typename T>
vector<T> fps_inv(const vector<T> &a,int deg=-1){
  int n=a.size();
  if(deg==-1)deg=n;
  assert(a[0]!=T(0));
  vector<T> ret(deg);
  ret[0]=a[0].inv();
  for(int m=1;m<deg;m<<=1){
    vector<T> f(a.begin(),a.begin()+min(n,m*2));
    if(f.size()<m*2)f.resize(m*2,0);
    vector<T> g(ret);
    f.resize(m*2);
    dft(f);
    g.resize(m*2);
    dft(g);
    rep(i,m*2)f[i]*=g[i];
    idft(f);
    T inv=T::raw(m*2).inv();
    rep(i,m)f[i]=0;
    reps(i,m,m*2)f[i]*=inv;
    dft(f);
    rep(i,m*2)f[i]*=g[i];
    idft(f);
    rep(i,m*2)f[i]*=inv;
    reps(i,m,min(deg,m*2))ret[i]-=f[i];
  }
  return ret;
}
template<typename T>
vector<T> fps_diff(const vector<T>&a){
  int n=a.size();
  vector<T>b(max(0,n-1));
  reps(i,1,n)b[i-1]=a[i]*i;
  return b;
}
template<typename T>
vector<T> fps_integral(const vector<T>&a){
  int n=a.size();
  vector<T>b(n+1);
  b[0]=0;
  if(n)b[1]=1;
  reps(i,2,n+1)b[i]=-b[T::mod()%i]*(T::mod()/i);
  rep(i,n)b[i+1]*=a[i];
  return b;
}
template<typename T>
vector<T> fps_log(const vector<T>&a,int deg=-1){
  int n=a.size();
  if(deg==-1)deg=n;
  vector<T>b=fps_integral(ntt_convolution(fps_diff(a),fps_inv(a,deg)));
  return {b.begin(),b.begin()+deg};
}
template<typename T>
vector<T> fps_exp(const vector<T>&a,int deg=-1){
  assert(a.empty()||a[0]==0);
  int n=a.size();
  if(deg==-1)deg=n;
  vector<T>b{1};
  for(int m=1;m<deg;m*=2){
    vector<T>nxt=fps_log(b,m*2);
    rep(i,m*2)nxt[i]*=-1;
    rep(i,min(n,m*2))nxt[i]+=a[i];
    nxt[0]++;
    b=ntt_convolution(b,nxt);
    b.resize(m*2);
  }
  return {b.begin(),b.begin()+deg};
}
template<typename T>
vector<T> fps_pow(vector<T> a,ll k,int deg=-1){
  int n=a.size();
  if(deg==-1)deg=n;
  if(k==0){
    vector<T>ret(deg,0);
    ret[0]=1;
    return ret;
  }
  int of=0;
  while(a[of]==0&&of!=n)of++;
  if(of==n)return vector<T>(deg,0);
  if(of!=0&&k>=deg)return vector<T>(deg,0);
  if(of*k>=deg)return vector<T>(deg,0);
  a.erase(a.begin(),a.begin()+of);
  n=a.size();
  T a0=a[0];
  T inv=a[0].inv();
  rep(i,n)a[i]*=inv;
  vector<T>lg=fps_log(a,deg);
  rep(i,deg)lg[i]*=k;
  vector<T>ep=fps_exp(lg,deg);
  T pw=a0.pow(k);
  rep(i,deg)ep[i]*=pw;
  vector<T>ret(deg,0);
  reps(i,of*k,deg)ret[i]=ep[i-of*k];
  return ret;
}
template<typename T>
vector<T> fps_pow_sparse(const vector<T>&a,ll k,int deg=-1){
  int n=a.size();
  if(deg==-1)deg=n;
  vector<pair<int,T>>b;
  rep(i,n)if(a[i]!=0)b.emplace_back(i,a[i]);
  if(b.empty()){
    vector<T>ret(deg,0);
    if(k==0)ret[0]=1;
    return ret;
  }
  int of=b[0].first;
  T b0=b[0].second,inv=b0.inv();
  for(auto &i:b){
    i.first-=of;
    i.second*=inv;
  }
  b0=b0.pow(k);
  vector<T>ret(deg,0);
  ret[0]=1;
  rep(i,deg-1){
    T cu=0;
    for(auto [j,c]:b){
      if(i-j+1<0)continue;
      if(j==0)continue;
      cu-=c*ret[i-j+1]*(i-j+1);
      cu+=ret[i-j+1]*k*j*c;
    }
    ret[i+1]=cu/(i+1);
  }
  if(of){
    if(k>=deg||of*k>=deg)ret.assign(deg,0);
    else{
      ret.insert(ret.begin(),of*k,0);
      ret.resize(deg);
    }
  }
  for(auto &i:ret)i*=b0;
  return ret;
}
template<typename T>
vector<T> fps_div_sparse(vector<T> a,const vector<T>&b,int deg=-1){
  int n=a.size(),m=b.size();
  if(deg==-1)deg=n;
  a.resize(deg,0);
  assert(b[0]!=0);
  vector<pair<int,T>>c;
  reps(i,1,m)if(b[i]!=0)c.emplace_back(i,b[i]);
  T v=b[0].inv();
  rep(i,deg){
    for(auto [j,e]:c){
      if(j>i)break;
      a[i]-=a[i-j]*e;
    }
    a[i]*=v;
  }
  return a;
}
template<typename T>
inline vector<T> fps_inv_sparse(const vector<T>&a,int deg=-1){
  return fps_div_sparse({1},a,(deg==-1?a.size():deg));
}
template<typename T>
vector<T>fps_fode_sparse(const vector<T>&a,const vector<T>&b){
  assert(a.size()==b.size());
  assert(a[0]==1);
  int n=a.size();
  vector<pair<int,T>>c,d;
  reps(i,1,n)if(a[i]!=0)c.emplace_back(i,a[i]);
  rep(i,n)if(b[i]!=0)d.emplace_back(i,b[i]);
  vector<T>f(n),df(n-1);
  f[0]=1;
  vector<T>inv(n);
  rep(i,n-1){
    T v=0;
    for(auto [j,x]:c){
      if(j>i)break;
      v+=x*df[i-j];
    }
    for(auto [j,x]:d){
      if(j>i)break;
      v+=x*f[i-j];
    }
    df[i]=-v;
    if(i==0)inv[1]=1;
    else inv[i+1]=-inv[T::mod()%(i+1)]*(T::mod()/(i+1));
    f[i+1]=df[i]*inv[i+1];
  }
  return f;
}
template<typename T>
vector<T> fps_pow_sparse(const vector<T>&a,const vector<T>&b,ll k,ll l,int deg=-1){
  int n=a.size(),m=b.size();
  if(deg==-1)deg=n;
  assert(a[0]==1);
  vector<pair<int,T>>c,d;
  rep(i,n)if(a[i]!=0)c.emplace_back(i,a[i]);
  rep(i,m)if(b[i]!=0)d.emplace_back(i,b[i]);
  vector<T>f(deg,0),g(deg,0);
  for(auto [i,x]:c){
    for(auto [j,y]:d){
      if(i+j>=deg+1)continue;
      T z=x*y;
      if(i+j<deg)f[i+j]+=z;
      if(0<i+j&&i+j<=deg)g[i+j-1]-=z*(T(i)*k+T(j)*l);
    }
  }
  return fps_fode_sparse(f,g);
}
template<typename T>
T bostan_mori(vector<T> p,vector<T> q,ll k){
  int z=1;
  int n=p.size(),m=q.size();
  while(z<m*2)z*=2;
  vector<T>mq(z);
  p.resize(z,0),q.resize(z,0),mq.resize(z,0);
  while(k){
    rep(i,z)mq[i]=q[i]*(i&1?-1:1);
    dft(p),dft(q),dft(mq);
    rep(i,z)p[i]*=mq[i];
    rep(i,z)q[i]*=mq[i];
    idft(p),idft(q);
    T inv=T::raw(z).inv();
    rep(i,z)p[i]*=inv;
    rep(i,z)q[i]*=inv;
    vector<T>u(n),v(m);
    if(k&1)rep(i,n)u[i]=p[i*2+1];
    else rep(i,n)u[i]=p[i*2];
    rep(i,m)v[i]=q[i*2];
    u.resize(z,0),v.resize(z,0);
    swap(p,u),swap(q,v);
    k>>=1;
  }
  return p[0];
}
template<typename T>
vector<T> berlekamp_massey(const vector<T>&s){
  const int n=s.size();
  vector<T>b,c;
  b.reserve(n+1),c.reserve(n+1);
  b.emplace_back(1),c.emplace_back(1);
  T y=1;
  reps(i,1,n+1){
    int l=c.size(),m=b.size();
    T x=0;
    rep(j,l)x+=c[j]*s[i-l+j];
    b.emplace_back(0);
    ++m;
    if(x==0)continue;
    T f=x/y;
    if(l<m){
      auto c2=c;
      c.insert(c.begin(),m-l,0);
      rep(j,m)c[m-1-j]-=f*b[m-1-j];
      b=c2;
      y=x;
    }
    else{
      rep(j,m)c[l-1-j]-=f*b[m-1-j];
    }
  }
  reverse(all(c));
  return c;
}
template<typename T>
T kth_term(vector<T>p,ll k){
  vector<T>q=berlekamp_massey(p);
  if(T::mod()==998244353){
    p=ntt_convolution(p,q);
    p.resize(q.size()-1);
    return bostan_mori(p,q,k);
  }
  auto convolution_naive=[](vector<T>&x,const vector<T>&y)->void {
    vector<T>temp(x.size()+y.size()-1,0);
    rep(i,x.size())rep(j,y.size())temp[i+j]+=x[i]*y[j];
    swap(x,temp);
  };
  convolution_naive(p,q);
  int n=q.size();
  p.resize(n-1);
  vector<T>mq(n);
  while(k){
    rep(i,n)mq[i]=q[i]*(i&1?-1:1);
    convolution_naive(p,mq);
    convolution_naive(q,mq);
    vector<T>u(n-1),v(n);
    if(k&1)rep(i,n-1)u[i]=p[i*2+1];
    else rep(i,n-1)u[i]=p[i*2];
    rep(i,n)v[i]=q[i*2];
    swap(p,u),swap(q,v);
    k>>=1;
  }
  if(p.empty())return 0;
  else return p[0];
}
using mint=modint1000000007;
void SOLVE(){
  ll n,a,b,k;
  cin>>n>>a>>b>>k;
  n--;
  vector<mint>dp(200);
  dp[0]=a,dp[1]=b;
  reps(i,2,200){
    dp[i]=(dp[i-1]*dp[i-1]+k)/dp[i-2];
  }
  cout<<kth_term(dp,n).val()<<endl;
}
0