[Luogu && UVA] UVA10359 题解

UVA10359 题解

闲的没事用 Python 切高精题，看见这题正好可以交题解，那我就来交一发。

递推。

题意：有一大堆的矩形，分为两种：\(2\times 1\) 和 \(2\times 2\) 的，求用这些矩形拼成一个 \(2\times n\) 的大矩形有几种组合。

观察题面可以发现，对于一个 \(2 \times n\) 的大矩形，拼法有以下三种：

在 \(2(n-1)\) 的大矩形后增加一个 \(2\times 1\) 的小矩形；

在 \(2(n-2)\) 的大矩形后增加两个 \(2\times 1\) 的小矩形；

在 \(2(n-2)\) 的大矩形后增加一个 \(2\times 2\) 的小矩形；

由上述三点，可以得到：对于不同的 \(i\)，拼法 \(p_i\) 的递推式为：

\[p_i=\begin{cases}1,i\in\{0,1\}\\p_{i-1}+2p_{i-2},i\ge2\end{cases} \]

直接代入计算...就行了嘛？

注意样例，\(n=200\) 时，\(p_{200}=1071292029505993517027974728227441735014801995855195223534251\)，而且 \(n \in [0,250]\)，用 C++ 直接打肯定会爆。

于是 Python，启动！

a = [1, 1]
for i in range(2, 251):
    a.append(a[i - 1] + 2 * a[i - 2] )
while True:
    try:
        n = int(input())
        print(a[n])
    except: break;

此代码可以 AC。

由于 Python 自带高精，可以省去一大堆麻烦事。

如果不看是否会爆数据类型，上述代码就等价于：

#include <bits/stdc++.h>

using namespace std;

int a[] = {1, 1}; 

int n;

int main() {
	for (int i = 2; i <= 251; ++i) {
		a[i] = a[i - 1] + a[i - 2] * 2;
	}
	while(cin >> n) {
		cout << a[n] << '\n';
	}
	return 0;
}

有了递推式，我们就可以写出打表代码：

生成：

a = [1, 1]

for i in range(2, 251):
    a.append (a[i - 1] +  a[i - 2] * 2)
print("string a[] = {", end = "")

for i in range(0, 125):
    print("\"", end = "")
    print(a[i], end = "\",")

print("")

for i in range(125, 250):
    print("\"", end = "")
    print(a[i], end = "\",")

print("\"", end = "")
print(a[250], end = "\"}; ")

将输出结果扔到 C++ 代码里：

#include <bits/stdc++.h>

using namespace std;

string a[] = 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int n;

int main() {
	while(cin >> n) {
		cout << a[n] << '\n';
	}
	return 0;
}