HDOJ/HDU 1297 Children’s Queue(推导~大数)

Problem Description

There are many students in PHT School. One day, the headmaster whose name is PigHeader wanted all students stand in a line. He prescribed that girl can not be in single. In other words, either no girl in the queue or more than one girl stands side by side. The case n=4 (n is the number of children) is like

FFFF, FFFM, MFFF, FFMM, MFFM, MMFF, MMMM

Here F stands for a girl and M stands for a boy. The total number of queue satisfied the headmaster’s needs is 7. Can you make a program to find the total number of queue with n children?

Input

There are multiple cases in this problem and ended by the EOF. In each case, there is only one integer n means the number of children (1<=n<=1000)

Output

For each test case, there is only one integer means the number of queue satisfied the headmaster’s needs.

Sample Input

1

2

3

Sample Output

1

2

4

题意：

就是n个人，站成一排。

有一个要求，(F)女生不能单独一个人站在男生之间。

可以没有女生。

输出有多少种站法；

(不考虑人与人的不同，只考虑位置和男女区别)

（如果一排以MF结尾是不合法的）

分析：

假如n个人的站法为db[n];

由前面的推导出db[n]。

db[n-1]结尾添加一个M，是一定可以的。

db[n-2]结尾添加FF，也是一定可以的。

添加MF不可以，添加MM也是可以的（但是这个情况和db[n-1]中重复了），添加FM也是和db[n-1]+M重复了。

在不可以序列后面加上FF(MF不可以,加上FF),成为合法,

所以db[n-4]后面+MFFF可以, 其实加一个F也能构成合法，但是这种情况包含在db[n-2]（相当与+FF）里面；

所以递推方程式db[n] =db[n-1] + db[n-2] + db[n-4];

db[i] 中保存的都是合法序列数。

Java大数秒A~~~

import java.math.BigInteger;
import java.util.Scanner;
public class Main{
    static BigInteger db[] = new BigInteger[1001];
    public static void main(String[] args) {
        dabiao();
        Scanner sc = new Scanner(System.in);
        while(sc.hasNext()){
            int n =sc.nextInt();
            System.out.println(db[n]);
        }
    }
    private static void dabiao() {
        db[0]=new BigInteger("1");
        db[1]=new BigInteger("1");
        db[2]=new BigInteger("2");
        db[3]=new BigInteger("4");
        db[4]=new BigInteger("7");
        for(int i=5;i<db.length;i++){
            db[i]=db[i-1].add(db[i-2]).add(db[i-4]);
        }
    }
}