Problem Description

Read the program below carefully then answer the question.

pragma comment(linker, “/STACK:1024000000,1024000000”)

include

include

include

include

include

include

const int MAX=100000*2;

const int INF=1e9;

int main()

{

int n,m,ans,i;

while(scanf(“%d%d”,&n,&m)!=EOF)

{

ans=0;

for(i=1;i<=n;i++)

{

if(i&1)ans=(ans*2+1)%m;

else ans=ans*2%m;

}

printf(“%d\n”,ans);

}

return 0;

}

Input

Multi test cases，each line will contain two integers n and m. Process to end of file.

[Technical Specification]

1<=n, m <= 1000000000

Output

For each case，output an integer，represents the output of above program.

Sample Input

1 10 3 100

Sample Output

1 5

Source

BestCoder Round #8

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n很大,线性方法不行

第一个奇数是1,产生的1经过了n-1次乘法,最终是2^(n-1)

第二个奇数是3,产生的1经过了n-3次乘法,最终是2^(n-3)

……

因此最后就是2^(n - 1) + 2 ^ (n -3 ) + …..

可以分n奇偶考虑

奇数: 2^0 + 2 ^ 2 + …. + 2 ^ (n-1) -> 4^0 + 4^1 + …+ 4 ^(n-1)/2

偶数:2 * (4^0 + 4^1 + … + 4^(n -2) / 2)

->矩阵快速幂求出即可

/*************************************************************************
    > File Name: hdu4990.cpp
    > Author: ALex
    > Mail: [email protected]
    > Created Time: 2015年03月15日 星期日 20时32分47秒
 ************************************************************************/

#include <map>
#include <set>
#include <queue>
#include <stack>
#include <vector>
#include <cmath>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <iostream>
#include <algorithm>
using namespace std;

const double pi = acos(-1.0);
const int inf = 0x3f3f3f3f;
const double eps = 1e-15;
typedef long long LL;
typedef pair <int, int> PLL;

LL mod, n;

class MARTIX
{
    public:
        LL mat[5][5];
        MARTIX();
        MARTIX operator * (const MARTIX &b)const;
        MARTIX& operator = (const MARTIX &b);
}A;

MARTIX::MARTIX()
{
    memset (mat, 0, sizeof(mat));
}

MARTIX MARTIX :: operator * (const MARTIX &b)const
{
    MARTIX ret;
    for (int i = 0; i < 3; ++i)
    {
        for (int j = 0; j < 3; ++j)
        {
            for (int k = 0; k < 3; ++k)
            {
                ret.mat[i][j] += this -> mat[i][k] * b.mat[k][j];
                ret.mat[i][j] %= mod;
            }
        }
    }
    return ret;
}

MARTIX& MARTIX :: operator = (const MARTIX &b)
{
    for (int i = 0; i < 3; ++i)
    {
        for (int j = 0; j < 3; ++j)
        {
            this -> mat[i][j] = b.mat[i][j];
        }
    }
    return *this;
}

MARTIX fastpow(MARTIX ret, LL n)
{
    MARTIX ans;
    for (int i = 0; i < 3; ++i)
    {
        ans.mat[i][i] = 1;
    }
    while (n)
    {
        if (n & 1)
        {
            ans = ans * ret;
        }
        ret = ret * ret;
        n >>= 1;
    }
    return ans;
}

void Debug(MARTIX A)
{
    for (int i = 0; i < 3; ++i)
    {
        for (int j = 0; j < 3; ++j)
        {
            printf("%lld ", A.mat[i][j]);
        }
        printf("\n");
    }
}

int main ()
{
    while (~scanf("%lld%lld", &n, &mod))
    {
        A.mat[0][0] = A.mat[0][2] = 4 % mod;
        A.mat[0][1] = A.mat[2][2] = 1 % mod;
        MARTIX F;
        F.mat[0][0] = 4 % mod;
        F.mat[0][1] = 1 % mod;
        F.mat[0][2] = 5 % mod;
        if (n == 0 || n == 1 || n == 2)
        {
            printf("%lld\n", n % mod);
            continue;
        }
        if (n & 1)
        {
            int cnt = ((n - 1) >> 1);
            MARTIX ans = fastpow(A, cnt - 1);
            ans = F * ans;
            printf("%lld\n", ans.mat[0][2]);
        }
        else
        {
            MARTIX ans = fastpow(A, ((n - 2) >> 1) - 1);
            ans = F * ans;
            printf("%lld\n", ans.mat[0][2] * 2 % mod);
        }
    }
    return 0;
}