HDU 6030 Happy Necklace (矩阵快速幂)

题目链接：http://acm.hdu.edu.cn/showproblem.php?pid=6030

思路：可以推导出递推式f[i]=f[i-1]+f[i-3],用矩阵快速幂求解即可。

可以参考挑战程序设计竞赛第二版200页详细解释。

代码：

const int inf = 0x3f3f3f3f;
typedef vector<long long> vec;
typedef vector<vec> mat;

mat mul(mat &A, mat &B){
    mat C(A.size(), vec(B[0].size()));
    for(int i = 0; i < A.size(); i++){
        for(int k = 0; k < B.size(); k++){
            for(int j = 0; j < B[0].size(); j++){
                C[i][j] = (C[i][j] + A[i][k] * B[k][j]) % mod;
            }
        }
    }
    return C;
}

mat pow(mat A, ll n){
    mat B(A.size(), vec(A.size()));
    for(int i = 0; i < A.size(); i++){
        B[i][i] = 1;
    }
    while(n > 0){
        if(n & 1) B = mul(B, A);
        A = mul(A, A);
        n >>= 1; 
    }
    return B;
}

int main(){
    int t;
    scanf("%d", &t);
    while(t--){
        long long n;
        scanf("%lld", &n);
        mat A(3, vec(3));
        A[0][0] = 1; A[0][1] = 0; A[0][2] = 1;
        A[1][0] = 1; A[1][1] = 0; A[1][2] = 0;
        A[2][0] = 0; A[2][1] = 1; A[2][2] = 0;
        A = pow(A, n - 2);
        ll ans = (A[2][0] * 6 + A[2][1] * 4 + A[2][2] * 3) % mod;
        printf("%lld
", ans);
    }
}

题目：

Happy Necklace

Time Limit: 2000/1000 MS (Java/Others)    Memory Limit: 131072/131072 K (Java/Others)
Total Submission(s): 539    Accepted Submission(s): 236

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

 

Source

2017中国大学生程序设计竞赛 - 女生专场

 

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