!Берликамп (OEIS во плоти)

#include <bits/stdc++.h>
using namespace std;
#define pb push_back
typedef long long ll;
#define SZ 233333
const int MOD=1e9+7; //or any prime
ll qp(ll a,ll b)
{
    ll x=1; a%=MOD;
    while(b)
    {
        if(b&1) x=x*a%MOD;
        a=a*a%MOD; b>>=1;
    }
    return x;
}
namespace linear_seq {
inline vector<int> BM(vector<int> x)
{
    //ls: (shortest) relation sequence (after filling zeroes) so far
    //cur: current relation sequence
    vector<int> ls,cur;
    //lf: the position of ls (t')
    //ld: delta of ls (v')
    int lf,ld;
    for(int i=0;i<int(x.size());++i)
    {
        ll t=0;
        //evaluate at position i
        for(int j=0;j<int(cur.size());++j)
            t=(t+x[i-j-1]*(ll)cur[j])%MOD;
        if((t-x[i])%MOD==0) continue; //good so far
        //first non-zero position
        if(!cur.size())
        {
            cur.resize(i+1);
            lf=i; ld=(t-x[i])%MOD;
            continue;
        }
        //cur=cur-c/ld*(x[i]-t)
        ll k=-(x[i]-t)*qp(ld,MOD-2)%MOD/*1/ld*/;
        vector<int> c(i-lf-1); //add zeroes in front
        c.pb(k);
        for(int j=0;j<int(ls.size());++j)
            c.pb(-ls[j]*k%MOD);
        if(c.size()<cur.size()) c.resize(cur.size());
        for(int j=0;j<int(cur.size());++j)
            c[j]=(c[j]+cur[j])%MOD;
        //if cur is better than ls, change ls to cur
        if(i-lf+(int)ls.size()>=(int)cur.size())
            ls=cur,lf=i,ld=(t-x[i])%MOD;
        cur=c;
    }
    for(int i=0;i<int(cur.size());++i)
        cur[i]=(cur[i]%MOD+MOD)%MOD;
    return cur;
}
int m; //length of recurrence
//a: first terms
//h: relation
ll a[SZ],h[SZ],t_[SZ],s[SZ],t[SZ];
//calculate p*q mod f
inline void mull(ll*p,ll*q)
{
    for(int i=0;i<m+m;++i) t_[i]=0;
    for(int i=0;i<m;++i) if(p[i])
        for(int j=0;j<m;++j)
            t_[i+j]=(t_[i+j]+p[i]*q[j])%MOD;
    for(int i=m+m-1;i>=m;--i) if(t_[i])
        //miuns t_[i]x^{i-m}(x^m-\sum_{j=0}^{m-1} x^{m-j-1}h_j)
        for(int j=m-1;~j;--j)
            t_[i-j-1]=(t_[i-j-1]+t_[i]*h[j])%MOD;
    for(int i=0;i<m;++i) p[i]=t_[i];
}
inline ll calc(ll K)
{
    for(int i=m;~i;--i)
        s[i]=t[i]=0;
    //init
    s[0]=1; if(m!=1) t[1]=1; else t[0]=h[0];
    //binary-exponentiation
    while(K)
    {
        if(K&1) mull(s,t);
        mull(t,t); K>>=1;
    }
    ll su=0;
    for(int i=0;i<m;++i) su=(su+s[i]*a[i])%MOD;
    return (su%MOD+MOD)%MOD;
}
inline int work(vector<int> x,ll n)
{
    if(n<int(x.size())) return x[n];
    vector<int> v=BM(x); m=v.size(); if(!m) return 0;
    for(int i=0;i<m;++i) h[i]=v[i],a[i]=x[i];
    return calc(n);
}
}
using linear_seq::work;
int main()
{
    vector<int> x = {1, 2, 4, 8, 16};
    for (int i = 0; i < 10; i++)
        cout << work(x, i) << endl;
}