82-珠子染色-置换群

http://poj.org/problem?id=1286

　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　Necklace of Beads

Time Limit: 1000MS		Memory Limit: 10000K
Total Submissions: 9655		Accepted: 3968

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

 

#include <iostream>
#include <algorithm>
#include <cmath> 
using namespace std;
#define LL long long

LL gcd(LL a, LL b){
	return b == 0 ? a : gcd(b, a % b);
}

int main(){
	LL n, sum = 0;
	while(cin >> n && n != -1){
		if(n == 0){
			cout << 0 << endl;
			continue;
		}
		//旋转变换 
		//n种置换，步长为x的置换的循环节有GCD(x,n)个
		for(int i = 1; i <= n; i++){
			sum += pow(3, gcd(i, n));
		}
		//翻转变换：要分奇偶
		if(n & 1){
			sum += n * pow(3, (n - 1) / 2 + 1);
		} 
		else{
			sum += n/2 * pow(3, (n - 2) / 2 + 2); //按两个点翻转 
			sum += n/2 * pow(3, n / 2); //按边翻转   
		}
		sum /= 2 * n;  //除以G：总的置换个数 
		cout << sum << endl; 
		sum = 0; 
	}
}