Find the number of different sequences with $2 n+1$ terms, where each term is $\pm 1$ and the absolute value of the sum of any odd number of consecutive terms does not exceed 1.

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Let $S_{j}$ denote the sum of the first $j$ terms of the sequence, with the convention that $S_{0}=0$.
We only need to consider the case $S_{1}=1$, as the case $S_{1}=-1$ is completely analogous. In this case, the problem is transformed into: finding the number of sequences $\left(S_{0}, S_{1}, \cdots, S_{2 n+1}\right)$ that satisfy $S_{0}=0$, $S_{1}=1$, $S_{j+1}=S_{j} \pm 1(j \geqslant 1)$, and for any $0 \leqslant k, m \leqslant n$, we have
$$
\left|S_{2 k}-S_{2 m+1}\right| \leqslant 1
$$

By setting $m=0$ in equation (1), it is easy to see that $S_{2 k}=0$ or 2; by setting $k=0$ in equation (1), it is easy to see that $S_{2 m+1}=-1$ or 1.
We now consider two cases:
(1) For each $1 \leqslant t \leqslant n$, if $S_{2 t+1}=1$, there are $2^{n}$ such sequences.
(2) If there exists $t(1 \leqslant t \leqslant n)$ such that $S_{2 t+1}=-1$, there are $2^{n}-1$ such sequences.
Thus, the number of sequences satisfying the conditions with $S_{1}=1$ is $2^{n}+2^{n}-1=2^{n+1}-1$.
Therefore, the total number of sequences satisfying the conditions is $2\left(2^{n+1}-1\right)=2^{n+2}-2$.