Problem 104

Pandigital Fibonacci ends

The Fibonacci sequence is defined by the recurrence relation:
$F_n=F_{n-1}+F_{n-2}$, where $F_1=1$ and $F_2=1$.

It turns out that $F_{541}$, which contains $113$ digits, is the first Fibonacci number for which the last nine digits are $1$-$9$ pandigital (contain all the digits $1$ to $9$, but not necessarily in order). And $F_{2749}$, which contains $575$ digits, is the first Fibonacci number for which the first nine digits are $1$-$9$ pandigital.

Given that $F_k$ is the first Fibonacci number for which the first nine digits AND the last nine digits are $1$-$9$ pandigital, find $k$.

两端全数字的斐波那契数

斐波那契数列由如下递归关系生成：
$F_n=F_{n-1}+F_{n-2}$，其中$F_1=1$，且$F_2=1$。

包含有$113$位数字的$F_{541}$是第一个最后$9$位数字是$1$至$9$全数字（包含$1$至$9$所有的数字，但不一定按照顺序）的斐波那契数，而包含有$575$位数字的$F_{2749}$是第一个前$9$位数字是$1$至$9$全数字的斐波那契数。

若$F_k$是第一个前$9$位数字和后$9$位数字都是$1$至$9$全数字的斐波那契数，求$k$。