Question 136
Let $n$ be a natural number with $n \ge 2$. For any real numbers $a_1, a_2, \dots, a_n$, we denote $S_0 = 1$ and define $S_k$ by
$$
S_k = \sum_{1 \le i_1 \lt i_2 \lt \dots \lt i_k \le n} a_{i_1} a_{i_2} \dots a_{i_k},
$$
that is, the sum of all products of any $k$ numbers chosen among $a_1, a_2, \dots, a_n$, $k \in \{1, 2, \dots, n\}$.
Find the number of all $n$-tuples $(a_1, a_2, \dots, a_n)$ for which the following relation holds:
$$
(S_n - S_{n-2} + S_{n-4} - \dots)^2 + (S_{n-1} - S_{n-3} + S_{n-5} - \dots)^2 = 2^n S_n.
$$
$$
S_k = \sum_{1 \le i_1 \lt i_2 \lt \dots \lt i_k \le n} a_{i_1} a_{i_2} \dots a_{i_k},
$$
that is, the sum of all products of any $k$ numbers chosen among $a_1, a_2, \dots, a_n$, $k \in \{1, 2, \dots, n\}$.
Find the number of all $n$-tuples $(a_1, a_2, \dots, a_n)$ for which the following relation holds:
$$
(S_n - S_{n-2} + S_{n-4} - \dots)^2 + (S_{n-1} - S_{n-3} + S_{n-5} - \dots)^2 = 2^n S_n.
$$