Each integer appears once in difference sequence Does there exist an increasing sequence $a_1,a_2,\dots$ of positive integers such that both of the following are fulfilled:


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*Each positive integer appears exactly once among $a_2-a_1,a_3-a_2,\dots$

*For some $n$, each positive integer at least $n$ appears exactly once among $a_3-a_1,a_4-a_2,a_5-a_3,\dots$?
If we only require the first condition, we can take the sequence $1,2,4,7,11,\dots$, while if we only require the second condition, we can construct the sequence by induction, making sure that each difference appears once.

To fulfil the first condition, the differences $a_i-a_{i-1}$ must be distinct positive integers. Thus, to avoid repeating a difference, $$a_n\ge a_0+\frac{n(n+1)}2.$$ The second condition requires that all but finitely many positive integers appear among the $a_i-a_{i-2}$. But the above growth is too fast to enable that. Thus the second condition cannot be fulfilled.
