求斐波那契单词的第n个字符

定义【摘抄自Wiki】

Let  be "0" and be "01". Now  (the concatenation of the previous sequence and the one before that).

The infinite Fibonacci word is the limit 

We have:

    0

    01

    010

    01001

    01001010

    0100101001001

...

The first few elements of the infinite Fibonacci word are:

0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, ...

Problem:

求斐波那契单词的第n个字符。即写例如以下函数：

char f(index)

思路：

初始条件：f(0)=0, f(1)=1

斐波那契单词分为两部分，Sn-1和Sn-2。

如果index >= length(Sn-1)。那么f(index) = f(index-length(Sn-1))。

不断递减index直到index <= 1为止。那么就可以得出f(index)的值。

C++代码例如以下：

#include <vector>

#include <iostream>

#include <cassert>

char getCharInFibonacciWord(int index)

{

                 using namespace std;

                assert(index >=0);

                 // The 48th fibonacci number is already out of range of int.

                 static int fibs[48] = {1,2,0};

                 while( index > 1)

                {

                                 int i=0;

                                 for(int i=1; i<_countof(fibs); ++i)

                                {

                                                 if(fibs[i] == 0)

                                                                fibs[i] = fibs[i-1] + fibs[i-2];

                                                 if(fibs[i] > index)

                                                {

                                                                index -= fibs[i-1];

                                                                 break;

                                                }

                                }

                }

                 return (index == 0) ? '0' : '1';

}

int _tmain(int argc, _TCHAR* argv[])

{

                 for(int i=0; i<100; ++i)

                                std::cout << getCharInFibonacciWord(i);

                std::cout << std::endl;

                 return 0;

}