3. Given a positive integer $n \geqslant 2$. If the integer sequence $x_{0}, x_{1}$, $\cdots, x_{n}$ satisfies:
(1) For any $0 \leqslant i \leqslant n$, we have $\left|x_{i}\right| \leqslant n$;
(2) For any $0 \leqslant i<j \leqslant n$, we have $x_{i} \neq x_{j}$;
(3) For any $0 \leqslant i<j<k \leqslant n$, we have
$$
\begin{array}{l}
\max \left\{\left|x_{k}-x_{i}\right|,\left|x_{k}-x_{j}\right|\right\} \\
=\frac{1}{2}\left(\left|x_{i}-x_{j}\right|+\left|x_{j}-x_{k}\right|+\left|x_{k}-x_{i}\right|\right),
\end{array}
$$

then the sequence is called a "good sequence". Find the number of good sequences.
3. For $0 \leqslant i<j<k \leqslant n$, from condition (3) we get
$$
\begin{array}{l}
\max \left\{\left|x_{k}-x_{i}\right|,\left|x_{k}-x_{j}\right|\right\} \\
=\frac{1}{2}\left(\left|x_{i}-x_{j}\right|+\left|x_{j}-x_{k}\right|+\left|x_{k}-x_{i}\right|\right) \\
=\max \left\{x_{i}, x_{j}, x_{k}\right\}-\min \left\{x_{i}, x_{j}, x_{k}\right\} .
\end{array}
$$

Then $x_{k}$ is not between $x_{i}$ and $x_{j}$ on the number line, i.e.,
$$
x_{i}<x_{k}<x_{j} \text { or } x_{j}<x_{k}<x_{i}
$$

do not hold.
First, let $x_{0}, x_{1}, \cdots, x_{n}$ be a permutation of $n+1$ consecutive integers that satisfies condition (3). From the previous discussion, we know that for $k=1,2, \cdots, n$, $x_{0}, x_{1}, \cdots, x_{k}$ is a permutation of $k+1$ consecutive integers.

Place each permutation $x_{0}, x_{1}, \cdots, x_{n}$ that satisfies condition (3) on the number line in the order of their indices.

Let $x_{0}=i$. Then $x_{1}$ has two possible positions: $i-1$ or $i+1$. When $x_{0}, x_{1}, \cdots, x_{k-1}$ have already been placed on the number line, $x_{k}$ has two possible positions: $\min _{0 \leq i \leq k-1}\left\{x_{i}\right\}-1$ or $\max _{0 \leqslant i \leqslant k-1}\left\{x_{i}\right\}+1$. Therefore, when all $x_{0}, x_{1}, \cdots, x_{n}$ that satisfy condition (3) are placed on the number line, their relative positions have $2^{n}$ possible arrangements, i.e., for $n+1$ consecutive integers, the number of permutations that satisfy condition (3) is $2^{n}$.

Finally, select any $n+1$ numbers from $\mathbf{Z} \cap[-n, n]$, and number them as $0,1, \cdots, n$ according to their size. Treat them as $n+1$ consecutive integers, and we know that the number of permutations that satisfy condition (3) is $2^{n}$.

Therefore, the total number of permutations that satisfy conditions (1), (2), and (3) is $\mathrm{C}_{2 n+1}^{n+1} \cdot 2^{n}$.