hdu-1817-polya

Necklace of Beads

Time Limit: 3000/1000 MS (Java/Others)    Memory Limit: 32768/32768 K (Java/Others)
Total Submission(s): 1036    Accepted Submission(s): 373

Problem Description

Beads of red, blue or green colors are connected together into a circular necklace of n beads ( n < 40 ). If the repetitions that are produced by rotation around the center of the circular necklace or reflection to the axis of symmetry are all neglected, how many different forms of the necklace are there? 

 

Input

The input has several lines, and each line contains the input data n. 
-1 denotes the end of the input file. 

 

Output

The output should contain the output data: Number of different forms, in each line correspondent to the input data.

 

Sample Input

4 5 -1

 

Sample Output

21 39

 

Source

Xi'an 2002

　　　　三种颜色，对n个点围成的环着色，求不同的方案个数。

　　　　polya定理，置换一共有n*2个，n个是旋转，n个是翻转，翻转分奇偶讨论一下对称轴过点or不过点，计算出循环个数就好了。

　　　　

#include<iostream>
#include<cstring>
#include<cstdio>
#include<map>
#include<set>
#include<vector>
#include<algorithm>
#include<cmath> 
using namespace std;
#define LL long long 
#define PI acos(-1.0)
LL thr[66]={1};
int gcd(int a,int b){
    return b==0?a:gcd(b,a%b);
}
int main()
{
    LL n,i,j;
    for(i=1;i<=40;++i) thr[i]=thr[i-1]*3;
    while(cin>>n){
        if(n==-1)break;
        if(n==0){
            puts("0");
            continue;
        }
        LL ans=0;
        for(i=1;i<=n;++i){
            ans+=thr[gcd(i,n)];
        }
        if(n%2==0){
            ans=ans+thr[n/2]*(n/2);
            ans=ans+thr[n/2+1]*(n/2);
        }
        else{
            ans=ans+thr[n/2+1]*n;
        }
        ans=ans/2;
        ans=ans/n;
        cout<<ans<<endl;
    }
    return 0;
}