HDOJ 1028 Ignatius and the Princess III(递推)

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谙忆 发表于 2021/05/26 23:36:39 2021/05/26
【摘要】 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, w...

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


思路:
(i,j)(i>=j)代表的含义是i为n,j为划分的最大的数字。
边界:a(i,0) = a(i, 1) = a(0, i) = a(1, i) = 1;
i|j==0时,无论如何划分,结果为1;

当(i>=j)时,
划分为{j,{x1,x2…xi}},{x1,x2,…xi}的和为i-j,
{x1,x2,…xi}可能再次出现j,所以是(i-j)的j划分,所以划分个数为a(i-j,j);
划分个数还需要加上a(i,j-1)(累加前面的);

当(i < j)时,
a[i][j]就等于a[i][i];

import java.util.Scanner;

public class Main{ static int a[][] = new int[125][125]; public static void main(String[] args) { dabiao(); Scanner sc = new Scanner(System.in); while(sc.hasNext()){ int n = sc.nextInt(); System.out.println(a[n][n]); } } private static void dabiao() { for(int i=0;i<121;i++){ a[i][0]=1; a[i][1]=1; a[0][i]=1; a[1][i]=1; } for(int i=2;i<121;i++){ for(int j=2;j<121;j++){ if(j<=i){ a[i][j]=a[i][j-1]+a[i-j][j]; }else{ a[i][j]=a[i][i]; } } } }
}

  
 
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文章来源: chenhx.blog.csdn.net,作者:谙忆,版权归原作者所有,如需转载,请联系作者。

原文链接:chenhx.blog.csdn.net/article/details/50982710

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