HDU 1028.Ignatius and the Princess III【分析】【8月20】

Ignatius and the Princess III

Problem Description

"Well, it seems the first problem is too easy. I will let you know how foolish you are later." feng5166 says.

"The second problem is, given an positive integer N, we define an equation like this:
  N=a[1]+a[2]+a[3]+...+a[m];
  a[i]>0,1<=m<=N;
My question is how many different equations you can find for a given N.
For example, assume N is 4, we can find:
  4 = 4;
  4 = 3 + 1;
  4 = 2 + 2;
  4 = 2 + 1 + 1;
  4 = 1 + 1 + 1 + 1;
so the result is 5 when N is 4. Note that "4 = 3 + 1" and "4 = 1 + 3" is the same in this problem. Now, you do it!"

 

Input

The input contains several test cases. Each test case contains a positive integer N(1<=N<=120) which is mentioned above. The input is terminated by the end of file.

 

Output

For each test case, you have to output a line contains an integer P which indicate the different equations you have found.

 

Sample Input

   
   
   
   

    
    
    
    4
10
20
   
   
   
   

 

Sample Output

   
   
   
   

    
    
    
    5
42
627
   
   
   
   

 

给定n，n最多分成n份，有几种分法？重点是分析：（分析过程可参考博文）

f[i][j]表示把整数 i 拆成最多 j 个数字所具有的方法数。

那么if (i >ｊ)　f[i][j] = f[i-j][j] + f[i][j-1]; 意思就是如果i>j，那么有两种方式：一种是先把i里面分理处j个1，然后再把i-j拆成最多i-j个数字；另一种是把i拆分成最多j-1个数字。

if (i < j) f[i][j] = f[i][i]; 意思就是如果i<j，那么这种情况和把数字i最多拆成i个数字的是一样的。

if (i == j) f[i][j] = f[i][j-1] + 1; 意思就是如果i==j，那么可以把数字i拆分成j-1个数字，也可以把数字i拆分成i个1（这个就是那个1的意义）

另外，当i==1||j==1时，f[i][j]=1。

代码如下：

#include <cstdio>
int main(){
    int n;
    while(scanf("%d",&n)==1){
        int i,j,f[125][125]={0};
        for(i=0;i<=n;i++)
            f[i][1]=f[1][i]=1;
        for(i=2;i<=n;i++)
        for(j=1;j<=n;j++){
        if(i>j) f[i][j]=f[i-j][j]+f[i][j-1];
            else if(i==j) f[i][j]=1+f[i][j-1];
            else f[i][j]=f[i][i];
        }
        printf("%d\n",f[n][n]);
    }
    return 0;
}