Let a sequence $X = \{x_1, x_2, ..., x_n\}$ and $n = k \times l$ in which $k,l \in \mathbb{N}$. We have the following conditions:
$$A)\sum_{i=1}^{n} x_i = 0$$ $$B)\sum_{i=1}^{n} x_i^2 = 1$$
I was wondering if we could convert the following expression to a formula based on $n, k, l$ or anything else:
$$\sum_{i=1}^{l} (x_{k(i-1)+1} + x_{k(i-1)+2} + ... + x_{k(i-1)+k})^2$$ $i.e,$ in a simple word:
$(x_1 + x_2 + ... + x_k)^2 + (x_{k+1} + x_{k+2} + ... + x_{2k})^2 + ... + (x_{k(l-1)+1} + x_{k(l-1)+2} + ... + x_n)^2$
In general, I think it is not possible. But by having the conditions $A$ and $B$, I am not sure whether it is possible or not.