矩阵快速幂--HDU 6030 Happy Necklace

Problem Description

Little Q wants to buy a necklace for his girlfriend. Necklaces are single strings composed of multiple red and blue beads.

Little Q desperately wants to impress his girlfriend, he knows that she will like the necklace only if for every prime length continuous subsequence in the necklace, the number of red beads is not less than the number of blue beads.

Now Little Q wants to buy a necklace with exactly n beads. He wants to know the number of different necklaces that can make his girlfriend happy. Please write a program to help Little Q. Since the answer may be very large, please print the answer modulo 109+7.

Note: The necklace is a single string, {not a circle}.

Input

The first line of the input contains an integer T(1≤T≤10000), denoting the number of test cases.

For each test case, there is a single line containing an integer n(2≤n≤1018), denoting the number of beads on the necklace.

Output

For each test case, print a single line containing a single integer, denoting the answer modulo 109+7.

Sample Input

2

2

3

Sample Output

3

4

题意： 给出红蓝两种，然后排成一个字符串，要求在每一个长度为素数的区间里面是的r(red)的数量不小与b(blue)的数量;

思路：想象当n为2的时候的情况是 rr,rb,br,三种情况，当n为3的时候相当于在后面添加一个b或者r,会发现形成rr的情况是前面rr和br的和，形成br的情况是前面的rb，而形成rb的情况是前面的rr,不能有前面的br形成rb，因为在素数为3的时候不能形成brb；

所以你会发现这个针对的素数只是2和3；

根据递推，设数组a[],b[],c[]分别为后面两个字母为rr,br,rb的字符串的数量，那么可以得到递推式：

a[i] = a[i - 1] + c[i - 1];b[i] = a[i - 1];c[i] = b[i - 1];

而题中要求的是所有的字符串，即s[n] = a[n] + b[n] + c[n]；

可以得出s[i] = s[i - 1] + s[i - 3];

n的范围是10^18，那么只能用到矩阵快速幂： 可以推出最初的矩阵为

1 0 1
1 0 0
0 1 0

#include <bits/stdc++.h>
using namespace std;
const int mod=1e9+7;
typedef long long ll;
struct martix{
    ll mo[4][4];
    martix(){
        memset(mo,0,sizeof(mo));
    }
};
martix mul(martix a,martix b){
    martix c;
    for(int i=0;i<4;i++){
        for(int j=0;j<4;j++){
            for(int k=0;k<4;k++){
                c.mo[i][j]=(c.mo[i][j]+a.mo[i][k]*b.mo[k][j])%mod;
            }
        }
    }
    return c;
}
ll powmod(martix a,ll n){
    martix T;
    for(int i=0;i<4;i++){
        T.mo[i][i]=1;
    }
    while(n){
        if(n&1) T=mul(a,T);
        n>>=1;
        a=mul(a,a);
    }
    return (T.mo[0][0]*6+T.mo[0][1]*4+T.mo[0][2]*3)%mod;
}
int main()
{
    ll t;
    cin>>t;
    while(t--){
        ll n;
        cin>>n;
        int ans[5]={0,0,3,4,6};
        if(n<=4){
            cout<<ans[n]%mod<<endl;
        }
        else{
            martix q;
            q.mo[0][0]=q.mo[0][2]=q.mo[3][2]=q.mo[1][0]=q.mo[2][1]=q.mo[3][2]=1;
            cout<<powmod(q,n-4)%mod<<endl;
        }
    }
    return 0;
}