Problem 1424

1. Define the problem and constraints: We need to find the number of sequences $x_1, x_2, \ldots, x_{2n-1}, x_{2n}$ where $x_i \in \{-1, 1\}$ for each $i$, such that for any integers $k$ and $m$ with $1 \le k \le m \le n$, the inequality $\left|\sum_{i=2k-1}^{2m} x_i \right| \le 2$ holds.

2. Group the terms: Consider the sequence in pairs $(x_{2i-1}, x_{2i})$ for $i = 1, 2, \ldots, n$. Define $S_i = x_{2i-1} + x_{2i}$. The possible values for $S_i$ are:
- $S_i = 2$ if $x_{2i-1} = 1$ and $x_{2i} = 1$
- $S_i = -2$ if $x_{2i-1} = -1$ and $x_{2i} = -1$
- $S_i = 0$ if $x_{2i-1} = 1$ and $x_{2i} = -1$ or $x_{2i-1} = -1$ and $x_{2i} = 1$

3. Analyze the constraints: The condition $\left|\sum_{i=2k-1}^{2m} x_i \right| \le 2$ implies that the sum of any contiguous subsequence of pairs must be between -2 and 2. This means that the sequence of $S_i$ values must be such that the sum of any contiguous subsequence of $S_i$ values is between -2 and 2.

4. Count the valid sequences:
- If $S_i = 0$, there are 2 ways to choose $(x_{2i-1}, x_{2i})$.
- If $S_i = 2$ or $S_i = -2$, there is only 1 way to choose $(x_{2i-1}, x_{2i})$.

5. Formulate the sequence: The sequence of $S_i$ values must be such that the sum of any contiguous subsequence is between -2 and 2. This means that the sequence can have any number of 0s, but the non-zero values must alternate between 2 and -2.

6. Calculate the number of sequences:
- Let $k$ be the number of pairs where $S_i = 0$. There are $\binom{n}{k}$ ways to choose these positions.
- For each of these $k$ positions, there are $2^k$ ways to assign $(x_{2i-1}, x_{2i})$.
- For the remaining $n-k$ positions, the non-zero $S_i$ values must alternate between 2 and -2. There are 2 ways to start this alternation (starting with 2 or starting with -2).

7. **Sum over all possible values of $k$**:
$$\text{Total number of sequences} = 2 \sum_{k=0}^{n} \binom{n}{k} 2^k$$

8. Simplify the expression: Using the binomial theorem, we have:
$$\sum_{k=0}^{n} \binom{n}{k} 2^k = (1 + 2)^n = 3^n$$
Therefore, the total number of sequences is:
$$2 \cdot 3^n$$

9. Subtract the overcounted sequences: We need to subtract the sequences where all $S_i = 0$, which is $2^n$ sequences.

10. Final count:
$$\text{Total number of valid sequences} = 2 \cdot 3^n - 2^n$$