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I am not sure if it's new although it may be an easy consequence of some theorem or lemma.The result is as follows: By choosing a set of numbers between $0$ and $n$(for any $n$) picking each number at most once and summing them you can create any number between 0 and $n(n+1) / 2$ . The above result can be easily proven by induction.

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    $\begingroup$ This is a roundabout way of defining triangular numbers. $T_n = 1+2+3+...+n = n(n+1)/2$ which is typically proved inductively. Add up a subset of $\lbrace 1...n\rbrace$, you get a smaller number. $\endgroup$
    – Spencer
    Nov 16 '20 at 23:17
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    $\begingroup$ I do not think there is a special name for this but it is a special case of the subset sum problem: the subset sum set of an initial segment of natural numbers is another initial segment of natural numbers. $\endgroup$
    – Conifold
    Nov 17 '20 at 8:58

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