Before I make myself clear, I wish to say that I do not want to know a sequence like the binary sequence. If possible, I'd want a sequence smaller than the binary sequence, of polynomial growth rate (please avoid exponential growth rates of terms) - though the latter option need not necessarily be satisfied if not possible.
In other words, I mean to say, given a certain $k$ and a length $n$ for the sequence, we have to generate a sequence of $n$ terms whose $k$ terms' sum is always unique.
An extremely trivial example, $1, 2, 3, 4, 5$ is a sequence of $5$ terms, where no $k$ terms' sum (here $k=1$) are equal in any way, and $6, 16, 30, 48$ for $n=4, k=2$ and the $a$th term $=ak(a+k)$.