Jzzhu and Sequences CodeForces

题目链接:点我

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Jzzhu has invented a kind of sequences, they meet the following property:

You are given x and y, please calculate fn modulo 1000000007 (1e9 + 7).

Input

The first line contains two integers x and y (|x|, |y| ≤ 1e9). The second line contains a single integer n (1 ≤ n ≤ 2·1e9).

Output

Output a single integer representing fn modulo 1000000007 (1e9 + 7).

Example

Input2 33Output1Input0 -12Output1000000006

Note

In the first sample, f2 = f1 + f3, 3 = 2 + f3, f3 = 1.In the second sample, f2 =  - 1;  - 1 modulo (1e9 + 7) equals (1e9 + 6).

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题意:

计算递推式的第n项对1e9 + 7 取模.

思路:

矩阵快速幂, 由于题目所给的 n 很大,我们不能直接递推出来,那么我们可以用矩阵快速幂来加速这个递推过程,矩阵快速幂的原理其实和快速幂的原理是一样的,只是变成了矩阵乘法.对于递推式
具体矩阵请看代码.
代码:

#include<cstdio>#include<algorithm>#include<cstring>#include<iostream>#include<cmath>using namespace std;typedef long long LL;const int mod = 1e9 + 7;struct mat{    LL a[3][3];    mat(){memset(a, 0 ,sizeof(a));}    mat operator *( mat b){//重载矩阵乘法        mat c;        for(int  i = 1; i <= 2; ++i)        for(int k = 1; k <= 2; ++k){            if(a[i][k])//一个小小的优化,但有时候很有用.                for(int  j = 1; j <= 2; ++j){                c.a[i][j] += a[i][k] * b.a[k][j] ;                    c.a[i][j] =  (c.a[i][j] % mod + mod ) %mod;                }        }return c;    }};mat qpow(mat x, LL n){    mat ans;    ans.a[1][1] = ans.a[2][2] = 1;//单位矩阵,    while(n){        if(n & 1) ans = ans * x;        x = x * x;        n >>= 1;    }    return ans;}int main(){    LL x, y, n;    scanf("%I64d %I64d %I64d", &x, &y, &n);    x = (x % mod + mod) % mod;    y = (y % mod + mod) % mod;    if(n == 1){        printf("%I64d\n",(x + mod ) % mod);        return 0;    } if(n == 2){    printf("%I64d\n",(y + mod) % mod);    return 0;    } mat q;    q.a[1][1]= q.a[2][1] = 1;//递推矩阵    q.a[1][2] = -1;    mat ans ;    ans = qpow(q, n-2);    LL sum = ans.a[1][1] * y % mod + ans.a[1][2] * x % mod;    printf("%I64d\n", (sum% mod + mod) % mod);    return 0;}