hdu1028 Ignatius and the Princess III 递推
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Ignatius and the Princess III
Time Limit: 2000/1000 MS (Java/Others) Memory Limit: 65536/32768 K (Java/Others)Total Submission(s): 17872 Accepted Submission(s): 12532
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!"
"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
41020
Sample Output
542627
转载自:http://blog.sina.com.cn/s/blog_677a3eb30100kqnn.html
传送门
首先,我们引进一个小小概念来方便描述吧,record[n][m]是把自然数划划分成所有元素不大于m的分法,例如:
当n=4,m=1时,要求所有的元素都比m小,所以划分法只有1种:{1,1,1,1};
当n=4,m=2时,。。。。。。。。。。。。。。。。只有3种{1,1,1,1},{2,1,1},{2,2};
当n=4,m=3时,。。。。。。。。。。。。。。。。只有4种{1,1,1,1},{2,1,1},{2,2},{3,1};
当n=4,m=5时,。。。。。。。。。。。。。。。。只有5种{1,1,1,1},{2,1,1},{2,2},{3,1},{4};
从上面我们可以发现:当n==1||m==1时,只有一种分法;
当n<m时,由于分法不可能出现负数,所以record[n][m]=record[n][n];
当n==m时,那么就得分析是否要分出m这一个数,如果要分那就只有一种{m},要是不分,那就是把n分成不大于m-1的若干份;即record[n][n]=1+record[n][n-1];
当n>m时,那么就得分析是否要分出m这一个数,如果要分那就{{m},{x1,x2,x3..}}时n-m的分法record[n-m][m],要是不分,那就是把n分成不大于m-1的若干份;即record[n][n]=record[n-m][m]+record[n][m-1];
那么其递归式:
dp[i][j]将整数i划分为最大不超过j的
#include <cstdio>#include <cstring>#include <algorithm>#include <iostream>using namespace std;int main(){int N, dp[120 + 5][120 + 5];for (int i = 1; i <= 123; i++) dp[i][1] = dp[1][i] = 1;for (int i = 2; i <= 123; i++) {for (int j = 2; j <= 123; j++) {if (i == j) {dp[i][j] = 1 + dp[i][j - 1];}else if (i < j) {dp[i][j] = dp[i][i];}else if (i > j) {dp[i][j] = dp[i][j - 1] + dp[i - j][j];}}}while (~scanf("%d", &N)) printf("%d\n", dp[N][N]);return 0;}
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