[OscarCunningham]

Is there a sequence where the sequence and all its differences contain each positive integer once?

Something like

    1 3 9   26  66
     2 6  17  40
      4 11  23
       7  12
        5

Oh, here it is: https://oeis.org/A035313

[thaumasiotes]

> Oh, here it is: https://oeis.org/A035313

That sequence is not known to match what you asked for:

>> Conjecturally, every positive integer occurs in the sequence or one of its n-th differences, which would imply that the sequence and its n-th differences partition the positive integers.

For an intuition of why this might be hard to prove, note that you had to insert 7 into your structure before you inserted 5. In the general case, there might be a long waiting period before you're able to place some particular integer n. It might be infinitely long.