3. Find all integer sequences $a_{1}, a_{2}, \cdots, a_{n}$, such that:
(1) For any $1 \leqslant i \leqslant n$, we have
$$
1 \leqslant a_{i} \leqslant n ;
$$
(2) For any $1 \leqslant i, j \leqslant n$, we have
$$
\left|a_{i}-a_{j}\right|=|i-j| .
$$
3. The sequences that satisfy the conditions are only
$$
a_{i}=i(1 \leqslant i \leqslant n)
$$

and $a_{i}=n+1-i(1 \leqslant i \leqslant n)$.
According to condition (2), for $2 \leqslant i \leqslant n$,
$$
\left|a_{i}-a_{1}\right|=i-1 .
$$

Then $\left|a_{i}-a_{1}\right|$ has $n-1$ different values.
For any $1 \leqslant j \leqslant n$, the integer $a_{j}$ satisfies $1 \leqslant a_{i} \leqslant n$, so only when $a_{1}=1$ or $n$, $\left|a_{i}-a_{1}\right|$ can take $n-1$ different values.
Thus, when $a_{1}=1$, $a_{i}=i(2 \leqslant i \leqslant n)$;
when $a_{1}=n$, $a_{i}=n+1-i(2 \leqslant i \leqslant n)$.