A sequence of positive integers is Palindromic if it reads the same forward and backward. For example:
23 11 15 1 37 37 1 15 11 23
1 1 2 3 4 7 7 10 7 7 4 3 2 1 1
A Palindromic sequence is Unimodal Palindromic if the values do not decrease up to the middle value and then (since the sequence is palindromic) do not increase from the middle to the end For example, the first example sequence above is NOT Unimodal Palindromic
while the second example is.
A Unimodal Palindromic sequence is a Unimodal Palindromic Decomposition of an integer N, if the sum of the integers in the sequence is N. For example, all of the Unimodal Palindromic Decompositions of the first few integers are given below:
1: (1)
2: (2), (1 1)
3: (3), (1 1 1)
4: (4), (1 2 1), (2 2), (1 1 1 1)
5: (5), (1 3 1), (1 1 1 1 1)
6: (6), (1 4 1), (2 2 2), (1 1 2 1 1), (3 3),
(1 2 2 1), ( 1 1 1 1 1 1)
7: (7), (1 5 1), (2 3 2), (1 1 3 1 1), (1 1 1 1 1 1 1)
8: (8), (1 6 1), (2 4 2), (1 1 4 1 1), (1 2 2 2 1),
(1 1 1 2 1 1 1), ( 4 4), (1 3 3 1), (2 2 2 2),
(1 1 2 2 1 1), (1 1 1 1 1 1 1 1)
Write a program, which computes the number of Unimodal Palindromic Decompositions of an integer.
2 3 4 5 6 7 8 10 23 24 131 213 92 0樣例輸出
2 2 3 2 4 4 5 3 6 7 7 5 8 11 10 17 23 104 24 199 131 5010688 213 1055852590 92 331143提示
N < 250
DP:用f[i][j]表示左邊爲j的時候,把i分解爲單調回文數串的種類數
狀態轉移方程f[i][j] = sum{f[i-2*j][k]----->k>=j && i-2*j >= k}
注意用long long
#include<iostream>
#include<cstring>
using namespace std;
long long f[255][255];
int main()
{
int N;
memset(f,0,sizeof(f));
for(int i = 1; i <= 250; ++i)
{
f[i][i] = 1;
if(i % 2 == 0)
f[i][i/2] = 1;
for(int j = 1; i-2*j >= j; ++j)
{
for(int k = j; i-2*j>=k;++k)
f[i][j] += f[i-2*j][k];
}
}
while(cin >> N)
{
if(N == 0)
return 0;
long long sum = 0;
for(int i = 1; i <= N; ++i)
sum += f[N][i];
cout << N << ' ' << sum << endl;
}
system("pause");
return 0;
}