Let $ n$ be a natural number, and we consider the sequence $ a_1, a_2 \ldots , a_{2n}$ where  $ a_i \in (\minus{}1,0,1)$
If we make the sum of consecutive members of the sequence, starting from one with an odd index and finishing in one with and even index, the result is $ \le 2$ and $ \ge \minus{}2$
How many sequence are there satisfying this conditions?
To solve this problem, we need to count the number of sequences \( a_1, a_2, \ldots, a_{2n} \) where each \( a_i \in \{-1, 0, 1\} \) and the sum of any consecutive subsequence starting from an odd index and ending at an even index is between \(-2\) and \(2\).

1. **Define the problem in terms of sequences and sums:**
   - We need to consider sequences of length \(2n\).
   - Each element of the sequence can be \(-1\), \(0\), or \(1\).
   - The sum of any subsequence starting from an odd index and ending at an even index must be between \(-2\) and \(2\).

2. **Identify the possible sums for pairs:**
   - The possible sums for pairs \((a_i, a_{i+1})\) are:
     \[
     \{-2, -1, 0, 1, 2\}
     \]
   - There are three types of \(0\) (i.e., \((0,0)\), \((1,-1)\), \((-1,1)\)), two types of \(\pm 1\) (i.e., \((1,0)\), \((0,1)\), \((-1,0)\), \((0,-1)\)), and one type of \(\pm 2\) (i.e., \((1,1)\), \((-1,-1)\)).

3. **Define recursive functions:**
   - Let \( F(n) \) be the number of sequences of length \(2n\) that satisfy the conditions.
   - Let \( f(n) \) be the number of sequences of length \(n\) with entries in \( S' = S \setminus \{0\} \) (taking different types into account) so that all consecutive sums have absolute value \(\leq 2\).

4. **Define auxiliary functions:**
   - Let \( g(n) \) and \( h(n) \) be the number of sequences of length \(n\) with entries in \( S' \) so that the sums of the first \(r\) terms ( \(1 \leq r \leq n\) ) lie in \(\{-2, -1, 0\}\) and \(\{0, 1, 2\}\) respectively.
   - Let \( g'(n) \) and \( h'(n) \) be the number of sequences of length \(n\) with entries in \( S' \) so that the sums of the first \(r\) terms ( \(1 \leq r \leq n\) ) lie in \(\{-2, -1, 0, 1\}\) and \(\{-1, 0, 1, 2\}\) respectively.

5. **Set up the recursive relations:**
   \[
   f(n) = g(n-1) + h(n-1) + 2g'(n-1) + 2h'(n-1)
   \]
   \[
   g(n) = h(n-1) + 2g'(n-1), \quad h(n) = g(n-1) + 2h'(n-1)
   \]
   \[
   g'(n) = 2g(n-1) + 2h'(n-1) + h(n-1), \quad h'(n) = 2h(n-1) + 2g'(n-1) + g(n-1)
   \]

6. **Combine auxiliary functions:**
   - Let \( a(n) = g(n) + h(n) \) and \( b(n) = g'(n) + h'(n) \).
   \[
   f(n) = a(n-1) + 2b(n-1)
   \]
   \[
   a(n) = a(n-1) + 2b(n-1), \quad b(n) = 3a(n-1) + 2b(n-1)
   \]

7. **Compute recursively:**
   - Start with initial conditions and compute recursively to find \( F(n) \).

The final answer is \( \boxed{ F(n) } \).