A - Ignatius and the Princess III
Time Limit:1000MS Memory Limit:32768KB 64bit IO Format:%I64d & %I64u
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
4 10 20
Sample Output
5 42 627
看上去貌似是考母函数之类的一道题,但是到现在并没有掌握母函数的知识,在这里占个位,以后补充。
这题的意思是问,拆分一个整数一共有多少种方法。
我的做法是 DP。
可得到下面三个状态转移方程。
dp[i][j]表示为用最大不超过j的整数去合并为i.
if(i<j)
dp[i][j]=dp[i][i];
else if(i>j)
dp[i][j]=dp[i-j][j]+dp[i][j-1];
else
dp[i][j]=dp[i][j-1]+1;
先DP打表,然后输出即可。
#include <stdio.h>
#define N 125
int n,dp[N][N];
int main()
{
for(int i=1;i<=120;i++)
dp[i][1]=1;
for(int j=1;j<=120;j++)
dp[1][j]=1;
for(int i=1;i<=120;i++)
for(int j=1;j<=120;j++)
if(i<j)
dp[i][j]=dp[i][i];
else if(i>j)
dp[i][j]=dp[i-j][j]+dp[i][j-1];
else
dp[i][j]=dp[i][j-1]+1;
while(scanf("%d",&n)>0)
printf("%d\n",dp[n][n]);
return 0;
}
在DISCUSS里看到还有大牛用完全背包做的。
#include<cstdio>
#define N 125
int n,dp[N];
int main()
{
dp [0] = 1;
for(int i = 1;i<=120;i++)
for(int j = i;j<=120;j++)
dp[j]+=dp[j-i];
while(scanf("%d",&n)>0)
printf("%d\n",dp[n]);
}