1998 Putnam A4

Let $A_1=0$ and $A_2=1$. For $n>2$, the number $A_n$ is defined by

concatenating the decimal expansions of $A_{n-1}$ and $A_{n-2}$ from

left to right. For example $A_3=A_2 A_1=10$, $A_4=A_3 A_2 = 101$,

$A_5=A_4 A_3 = 10110$, and so forth. Determine all $n$ such that

$11$ divides $A_n$.

令$A_1=0$ 和$A_2=1$。对于 $n>2$，数字 $A_n$ 定义为

连接 $A_{n-1}$ 和 $A_{n-2}$ 的十进制扩展

从左到右。例如$A_3=A_2 A_1=10$，$A_4=A_3 A_2 = 101$，

$A_5=A_4 A_3 = 10110$，依此类推。确定所有 $n$ 使得

$11$ 除以 $A_n$。