Ignatius and the Princess III
Time Limit: 2000/1000 MS (Java/Others) Memory Limit: 65536/32768 K (Java/Others)Total Submission(s): 19924 Accepted Submission(s): 13936
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
4 10 20
Sample Output
5 42 627
Author
Ignatius.L
将一个正整数分成若干个数相加,求分解的方案数。
法一:动态规划:易得dp[i] = dp[i] + dp[i-a[k]], (k<=i)。
# include <stdio.h> int main() { int n, i, j, dp[125]={1}; for(i=1; i<=120; ++i) for(j=i; j<=120; ++j) dp[j] += dp[j-i]; while(~scanf("%d",&n)) printf("%d\n",dp[n]); return 0; }
法二:母函数:
# include <stdio.h> # define MAXN 120 int a[MAXN+3]={1}, b[MAXN+3]={0}; void init() { int i, j, k; for(i=1; i<=MAXN; ++i)//枚举分解的数 { for(j=0; j<=MAXN; j+=i)//从0开始,步长为i for(k=0; j+k<=MAXN; ++k) b[j+k] += a[k];//多项式的系数相加 for(j=0; j<=MAXN; ++j) { a[j] = b[j];//更新a数组 b[j] = 0;//清除b数组 } } } int main() { init(); int n; while(~scanf("%d",&n)) printf("%d\n",a[n]); return 0; }