2019.01.02 bzoj5300: [Cqoi2018]九连环（fft优化高精+快速幂）

传送门

题意不好描述（自己看样例解释）

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首先可以推出一个递推式：fn=fn−1+2fn−2+1f_n=f_{n-1}+2f_{n-2}+1fn​=fn−1​+2fn−2​+1

然后可以构造两个等式：

1. (fn+fn−1+1)=2(fn−1+fn−2+1)(f_n+f_{n-1}+1)=2(f_{n-1}+f_{n-2}+1)(fn​+fn−1​+1)=2(fn−1​+fn−2​+1)
2. (fn−2fn−1−12)=−(fn−1−fn−12)(f_n-2f_{n-1}-\frac12)=-(f_{n-1}-f_n-\frac12)(fn​−2fn−1​−21​)=−(fn−1​−fn​−21​)

然后变形一波可以得到fn=⌊2n+1−13⌋f_n=\left\lfloor\frac{2^{n+1}-1}3\right\rfloorfn​=⌊32n+1−1​⌋

接着我们惊奇的发现没有模数。

只能fftfftfft加快速幂了。

代码：

#include<bits/stdc++.h>
#define ri register int
using namespace std;
typedef long double ldb;
struct Cp{
    ldb x,y;
    friend inline Cp operator+(const Cp&a,const Cp&b){return (Cp){a.x+b.x,a.y+b.y};}
    friend inline Cp operator-(const Cp&a,const Cp&b){return (Cp){a.x-b.x,a.y-b.y};}
    friend inline Cp operator*(const Cp&a,const Cp&b){return (Cp){a.x*b.x-a.y*b.y,a.x*b.y+a.y*b.x};}
    friend inline Cp operator*(const Cp&a,const ldb&b){return (Cp){a.x*b,a.y*b};}
    friend inline Cp operator/(const Cp&a,const ldb&b){return (Cp){a.x/b,a.y/b};}
};
const ldb pi=acos(-1.0);
int lim,tim;
vector<Cp>A,B;
vector<int>pos;
inline void init(const int&up){
    lim=1,tim=0;
    while(lim<=up)lim<<=1,++tim;
    pos.resize(lim),A.resize(lim),B.resize(lim),pos[0]=0;
    for(ri i=0;i<lim;++i)pos[i]=(pos[i>>1]>>1)|((i&1)<<(tim-1));
}
inline void fft(vector<Cp>&a,const int&type){
    for(ri i=0;i<lim;++i)if(i<pos[i])swap(a[i],a[pos[i]]);
    for(ri mid=1;mid<lim;mid<<=1){
        Cp wn=(Cp){cos(pi/mid),type*sin(pi/mid)};
        for(ri j=0,len=mid<<1;j<lim;j+=len){
            Cp w=(Cp){1,0},a0,a1;
            for(ri k=0;k<mid;++k,w=w*wn){
                a0=a[j+k],a1=a[j+k+mid]*w;
                a[j+k]=a0+a1,a[j+k+mid]=a0-a1;
            }
        }
    }
    if(type==-1)for(ri i=0;i<lim;++i)a[i]=a[i]/(ldb)lim;
}
struct poly{
    vector<Cp>a;
    poly(int k=0,Cp x=(Cp){0,0}){a.resize(k+1),a[k]=x;}
    inline int deg()const{return a.size()-1;}
    inline poly extend(const int&k){poly ret=*this;return ret.a.resize(k+1),ret;}
    inline Cp&operator[](const int&k){return a[k];}
    inline const Cp&operator[](const int&k)const{return a[k];}
    friend inline poly operator*(const poly&a,const poly&b){
        int n=a.deg(),m=b.deg();
        init(n+m);
        poly ret(lim);
        for(ri i=0;i<=n;++i)A[i]=a[i];
        for(ri i=0;i<=m;++i)B[i]=b[i];
        for(ri i=n+1;i<lim;++i)A[i]=(Cp){0,0};
        for(ri i=m+1;i<lim;++i)B[i]=(Cp){0,0};
        fft(A,1),fft(B,1);
        for(ri i=0;i<lim;++i)A[i]=A[i]*B[i];
        return fft(A,-1),ret.a=A,ret;
    }
    friend inline poly operator*(const poly&a,const ldb&b){
        poly ret(a.deg());
        for(ri i=0;i<=a.deg();++i)ret[i]=a[i]*b;
        return ret;
    }
    friend inline poly operator/(const poly&a,const ldb&b){
        poly ret(a.deg());
        for(ri i=0;i<=a.deg();++i)ret[i]=a[i]/b;
        return ret;
    }
};
inline poly update(const poly&a){
    vector<int>res;
    int sum=0;
    for(ri val,i=0,up=a.deg();i<=up;++i)val=((int)(a[i].x+0.5))+sum,sum=val/10,res.push_back(val%10);
    while(sum)res.push_back(sum%10),sum/=10;
    while(!res[res.size()-1]&&res.size()!=1)res.pop_back();
    poly ret(res.size()-1);
    for(ri i=0;i<=ret.deg();++i)ret[i]=(Cp){res[i],0};
    return ret;
}
inline void ksm(int p){
    poly ret,a;
    ret[0]=(Cp){1,0},a[0]=(Cp){2,0};
    while(p){
        a=update(a);
        if(p&1)ret=update(ret),ret=ret*a;
        a=a*a,p>>=1;
    }
    ret=update(ret),reverse(ret.a.begin(),ret.a.end());
    vector<int>ans;
    for(ri val,divv=0,i=0;i<=ret.deg();++i)val=(int)ret[i].x+divv,ans.push_back(val/3),divv=val%3*10;
    int pos=0;
    while(!ans[pos]&&pos!=ans.size()-1)++pos;
    for(ri i=pos;i<ans.size();++i)cout<<ans[i];
    puts("");
}
char s[10005];
int main(){
    int n,m;
    scanf("%d",&m);
    while(m--)scanf("%d",&n),ksm(n+1);
    return 0;
}