I'm looking for vectors of $n$ numbers that carry the property that the unique sums of elements of the components of the vector is small.
For example for any $n$ the consider the vector $(1,1,...1)$ consisting of $n$ contiguous $1's$. Interpreting the vector as a set of elements the set of possible sums of these elements is $0,1,2,3,4...n$. I'm looking for other examples of classes of vectors (up to a multiplicative factor of a positive real number) where the possible sums of elements is at most $n+1$ but the vectors contain $n$ non-zero, REAL components.
Any vector of the form $(\pm 1, \pm 1, \pm 1, ... \pm 1)$ (where each component is a positive or negative 1 selected independently) is of the this form.
And it's not clear to me if any other vectors exist that carry my desired property and are not a positive multiple of one of the $2^n$ vectors in the aforementioned set.