linear_recurrence.cpp

#include <stdio.h>
#include <algorithm>
#include <vector>
#include <array>
#include <queue>
#include <deque>
#include <set>
#include <map>
#include <stdlib.h>
#include <ctime>
#include <climits>
#include <cmath>
#include <complex>
#include <iostream>
#include <cctype>
#include <cstring>
#include <numeric>
#include <bitset>
#include <stack>
#include <functional>
#include <cassert>
#include <tuple>
#include <iomanip>
#define pb push_back
#define mp make_pair
#define all(a) begin(a),end(a)
#define FOR(x,val,to) for(int x=(val);x<int((to));++x)
#define FORE(x,val,to) for(auto x=(val);x<=(to);++x)
#define FORR(x,arr) for(auto &x: arr)
#define FORS(x,plus,arr) for(auto x = begin(arr)+(plus); x != end(arr); ++x)
#define FORREV(x,plus,arr) for(auto x = (arr).rbegin()+(plus); x !=(arr).rend(); ++x)
#define REE(s_) {cout<<s_<<'\n';exit(0);}
#define GET(arr) for(auto &i: (arr)) sc(i)
#define whatis(x) cerr << #x << " is " << (x) << endl;
#define e1 first
#define e2 second
#define INF 0x7f7f7f7f
typedef std::pair<int,int> pi;
typedef std::vector<int> vi;
typedef std::vector<std::string> vs;
typedef int64_t ll;
typedef uint64_t ull;
#define umap unordered_map
#define uset unordered_set
using namespace std;

#ifdef ONLINE_JUDGE
#define whatis(x) ;
#endif
#ifdef _WIN32
#define getchar_unlocked() _getchar_nolock()
#define _CRT_DISABLE_PERFCRIT_LOCKS
#endif
template<class L, class R> ostream& operator<<(ostream &os, map<L, R> P) { for(auto const &vv: P)os<<"("<<vv.first<<","<<vv.second<<")"; return os; }
template<class T> ostream& operator<<(ostream &os, set<T> V) { os<<"[";for(auto const &vv:V)os<<vv<<","; os<<"]"; return os; }
template<class T> ostream& operator<<(ostream &os, vector<T> V) { os<<"[";for(auto const &vv:V)os<<vv<<","; os<<"]"; return os; }
template<class L, class R> ostream& operator<<(ostream &os, pair<L, R> P) { os<<"("<<P.first<<","<<P.second<<")"; return os; }
inline int fstoi(const string &str){auto it=str.begin();bool neg=0;int num=0;if(*it=='-')neg=1;else num=*it-'0';++it;while(it<str.end()) num=num*10+(*it++-'0');if(neg)num*=-1;return num;}
inline void getch(char &x){while(x = getchar_unlocked(), x < 33){;}}
inline void getstr(string &str){str.clear(); char cur;while(cur=getchar_unlocked(),cur<33){;}while(cur>32){str+=cur;cur=getchar_unlocked();}}
template<typename T> inline bool sc(T &num){ bool neg=0; int c; num=0; while(c=getchar_unlocked(),c<33){if(c == EOF) return false;} if(c=='-'){ neg=1; c=getchar_unlocked(); } for(;c>47;c=getchar_unlocked()) num=num*10+c-48; if(neg) num*=-1; return true;}template<typename T, typename ...Args> inline void sc(T &num, Args &...args){ bool neg=0; int c; num=0; while(c=getchar_unlocked(),c<33){;} if(c=='-'){ neg=1; c=getchar_unlocked(); } for(;c>47;c=getchar_unlocked()) num=num*10+c-48; if(neg) num*=-1; sc(args...); }
#define N 1000001

// Berlekamp-Massey algorithm.
#define SZ 233333
constexpr 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);
}
}

int main(){
    ios_base::sync_with_stdio(0);cin.tie(0);
	cout << linear_seq::work({1,1,2,3,5,8,13,21}, 10) << '\n';
}