Number of possible values for a sum of $n$ variables Let $x_k\in \{2^k,-2^k\}$ where $k\in \{0,1,\ldots,n-1\}$.

I am trying to check if the possible values for $y:=x_0+x_1+\ldots +x_{n-1}$ are unique?

I was trying to prove that they are indeed unique by induction. If $y=x_0$, they are unique. By induction hypothesis, let the possible values for $y=x_0+x_1+\ldots +x_k$ be unique but am not sure how to apply that for the sum of $k+1$ terms. Any ideas for proving or disproving this?
 A: Induction is one technique that is common to try out. I don't know that that pans out very well here. Consider what would happen to the (hypothetical) first $n$ that allows non-uniqueness. You would have a $y$ that is written in two different ways where the $x_{n-1}$ are necessarily distinct. What can the induction hypothesis say about the remaining $x_i$? Not much, as they don't sum to the same value: one set sums to $y-2^{n-1}$ and the other set sums to $y+2^{n-1}$. I'm not saying it can't be done, but it doesn't feel natural.
Another approach, that is often suited to proofs of uniqueness of things, is to study the difference between two potentially distinct things. (Not necessarily the subtraction between the things, although that works often enough.)
Take a number $y$ that can be written in two such ways, and subtract those two ways, term-by-term. Now you have $0$ written on the form
$$
0 = a_0 + a_1 + a_2 + \cdots + a_{n-1}
$$
where $a_k \in \{-2\cdot2^k, 0, 2\cdot 2^k\} = \{0, \pm 2^{k+1}\}$. For contradiction, let $i$ be the smallest index such that $a_i\neq 0$. Now consider whether the right-hand side of the equality above is divisible by $2^{i+2}$.
