UVA 10198 Counting
Counting
Counting |
The Problem
Gustavo knows how to count, but he is now learning how write numbers. As he is a very good student, he already learned 1, 2, 3 and 4. But he didn't realize yet that 4 is different than 1, so he thinks that 4 is another way to write 1. Besides that, he is having fun with a little game he created himself: he make numbers (with those four digits) and sum their values. For instance:
132 = 1 + 3 + 2 = 6 112314 = 1 + 1 + 2 + 3 + 1 + 1 = 9 (remember that Gustavo thinks that 4 = 1)After making a lot of numbers in this way, Gustavo now wants to know how much numbers he can create such that their sum is a number n. For instance, for n = 2 he noticed that he can make 5 numbers: 11, 14, 41, 44 and 2 (he knows how to count them up, but he doesn't know how to write five). However, he can't figure it out for n greater than 2. So, he asked you to help him.
The Input
Input will consist on an arbitrary number of sets. Each set will consist on an integer n such that 1 <= n <= 1000. You must read until you reach the end of file.
The Output
For each number read, you must output another number (on a line alone) stating how much numbers Gustavo can make such that the sum of their digits is equal to the given number.
Sample Input
2 3
Sample Output
5 13
题意:Gustavo数数时总是把1和4搞混,他觉得4仅仅是1的第二种写法。给出一个整数n,Gustavo想知道有多少个数的数字之和恰好为n。比如,当n=2时,有5个数:11、14、41、44、2。
分析:如果 F(n) 表示使用 1。2,3,4 构建的和为 n 的序列总数,则这些序列中,以 1 为開始的序列种数为 F(n - 1)。以2为開始的为 F(n - 2)。以3開始的序列总数为 F(n - 3)、以4開始的序列总数为 F(n - 4),因为 Gustavo 把 4 当作 1,则有 F(n - 4) = F(n - 1),
故 F(n) = F(n - 1) + F(n - 2) + F(n - 3) + F(n - 4) = 2 * F(n - 1) + F(n - 2) + F(n - 3)。
边界条件: F(1) = 2, F(2) = 5。 F(3) = 13。
故 F(n) = F(n - 1) + F(n - 2) + F(n - 3) + F(n - 4) = 2 * F(n - 1) + F(n - 2) + F(n - 3)。
边界条件: F(1) = 2, F(2) = 5。 F(3) = 13。
#include<string> #include<iostream> #include<vector> #include<algorithm> using namespace std; vector<string> v; string add(string a, string b) { string s; reverse(a.begin(), a.end()); reverse(b.begin(), b.end()); int i = 0; int m, k = 0; while(a[i] && b[i]) { m = a[i] - '0' + b[i] - '0' + k; k = m / 10; s += (m % 10 + '0'); i++; } if(i == a.size()) { while(i != b.size()) { m = k + b[i] - '0'; k = m / 10; s += m % 10 + '0'; i++; } if(k) s += k + '0'; } else if(i == b.size()) { while(i != a.size()) { m = k + a[i] - '0'; k = m / 10; s += m % 10 + '0'; i++; } if(k) s += k + '0'; } reverse(s.begin(), s.end()); return s; } void solve() { v.push_back("0"); v.push_back("2"); v.push_back("5"); v.push_back("13"); string s; for(int i = 4; ; i++) { s = add(v[i-1], v[i-1]); s = add(v[i-2], s); s = add(v[i-3], s); v.push_back(s); if(v[i].size() > 1001) break; } } int main() { solve(); int n; int Size = v.size(); while(cin >> n) { cout << v[n] << endl; } return 0; }
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