I would need a Sidon sequence of about $10^9$ elements. I found math papers like  that explain how to generate Sidon sequences but it seems a lot of pain to write the corresponding program.
Are you aware of any existing program that generates a dense enough and large enough Sidon sequence ?
Extra: my real constraint for the sequence is $a+a′ , a \lt a′ ,~ a,a′ ∈ A$ instead of the true Sidon sequence property $a+a ′ , a ≤ a ′ , a, a ′ ∈ A$. So a Sidon sequence would do the job but I could hope for a denser sequence by tuning the program.
 Javier Cilleruelo, "Infinite Sidon sequences", eprint on arXiv: http://arxiv.org/abs/1209.0326