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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 $=ak(a+k)$.

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For the case $k=2$, you may be interested in the Mian-Chowla sequence, seen at http://oeis.org/A005282. It starts 1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, 123, 148, 182, 204, 252, 290, 361, 401, 475, 565, 593, 662, 775, 822, 916, 970, 1016, 1159, 1312, 1395, 1523, 1572, 1821, 1896, 2029, 2254, 2379, 2510, 2780, 2925, 3155, 3354, 3591, 3797, 3998, 4297, 4433, 4779, 4851, and each term is as small as possible given the restriction of $a+b\ne c+d$. You might be interested in $B_2$ sequences more generally. There is a copious literature.

EDIT: Guy, 3rd edition, page 351 (problem E28) writes, "Let $a_1\lt a_2\lt\cdots$ be an infinite sequence of integers for which all the triple sums $a_i+a_j+a_k$ are distinct. Erdos offers \$500.00 for a proof or disproof of an old conjecture of his, that $\lim(a_n/n^3)=\infty$."

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    No, e.g., in Mian-Chowla, $1+4+13=2+8+8$, $2+4+81=8+13+66$, etc. For $k=2$ it was easy to find the first five or so terms by hand, then search the OEIS.2012-04-17