欧拉工程第65题：Convergents of e

题目链接

 

现在做这个题目真是千万只草泥马在心中路过

 

这个与上面一题差不多

 

这个题目是求e的第100个分数表达式中分子的各位数之和

 

What is most surprising is that the important mathematical constant,
e = [2; 1,2,1, 1,4,1, 1,6,1 , ... , 1,2k,1, ...].

The first ten terms in the sequence of convergents for e are:

2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, 1264/465, 1457/536, ...

The sum of digits in the numerator of the 10^th convergent is 1+4+5+7=17.

Find the sum of digits in the numerator of the 100^th convergent of the continued fraction for e.

 

上面可以发现一个规律

 

维基百科链接：连分数，上面有递推公式

 

 

这么多就足够解题了，

 

  上面的定理1，用不到的

 

e = [2; 1,2,1, 1,4,1, 1,6,1 , ... , 1,2k,1, ...].

a0=2

下面的：1,2,1,1,4,1,1,6,1，这个规律很明显

 

 

 

 

根据上面的规律

 

Java代码：

package project61;

import java.math.BigInteger;

public class P65{
    void run(){
        BigInteger d = new BigInteger("1");
        BigInteger n = new BigInteger("2");
        for(int i= 2;i<=100;i++){
            BigInteger temp = d;
            long c = (i%3==0)?2*(i/3):1;
            BigInteger BigC = new BigInteger(c+"");
            d = n;
            n = d.multiply(BigC).add(temp);
        }
        String toStr = n.toString();
        int result = 0;
        for(int i=0;i<toStr.length();i++){
            result += Integer.valueOf(toStr.charAt(i)+"");
        }
        System.out.println(toStr+"\nresult:"+result);
    }
    public static void main(String[] args){
        long start = System.currentTimeMillis();
        new P65().run();
        long end = System.currentTimeMillis();
        long time = end - start;
        System.out.println("run time:"+time/1000+"s"+time%1000+"ms");
    }
}

 

如果刚看到这一题的时候应该感觉这个数不是很大，然而分子是：6963524437876961749120273824619538346438023188214475670667

分子各位的数字和是：272，要用BigInteger类型

 

Python程序：

import time as time 

def mysum(num):
    return sum(map(int,str(num)))

def getA(i):
    if i%3==0:
        return 2*(i/3)
    else:
        return 1 
    
def run():
    h0 = 1
    h1 = 2
    for i in range(2,101):
        a = getA(i)
        h2 = a * h1 + h0 
        h0 = h1
        h1 = h2
    return mysum(h2)
    
if __name__== '__main__':
    start = time.time()
    result = run()
    print "running time={0},result={1}".format((time.time()-start),result)

   

Python是根据上面截图中的写的

running time=0.0,result=272