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 $k$ 。