POJ 1286 Necklace of Beads

Necklace of Beads

Time Limit: 1000MS		Memory Limit: 10000K
Total Submissions: 6061		Accepted: 2515

Description

Beads of red, blue or green colors are connected together into a circular necklace of n beads ( n < 24 ). 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
    很经典的一道题目，用Polya定理或者Burnside引理解决

#include <iostream>
#include <cstring>
#include <cstdio>
#include <cmath>
#define N 30
using namespace std;
__int64 a[N];
int main()
{
    //freopen("data.in","r",stdin);
    int gcd(int x,int y);
    a[1] = 3;
    for(int i=2;i<=25;i++)
    {
        a[i]=a[i-1]*3;
    }
    int n;
    while(scanf("%d",&n)!=EOF)
    {
        if(n==-1)
        {
            break;
        }
        if(n==0)
        {
            printf("0\n");
            continue;
        }
        __int64 res1=0;
        for(int i=1;i<=n;i++)
        {
            res1+=(a[gcd(i,n)]);
        }
        __int64 res2=0;
        if(n%2)
        {
            res2 +=(n*a[(n+1)/2]);
        }else
        {
            res2+=((n/2)*(a[(n/2)+1]+a[n/2]));
        }
        cout<<(res1+res2)/2/n<<endl;
    }
    return 0;
}
int gcd(int x,int y)
{
    return y==0 ? x:gcd(y,x%y);
}