I think it depends on the social setting. For instance, if you are in a class with someone, you can basically plan on seeing them for the entire semester, so I would give it time and not rush to get contact info.
Compared to, say, meeting someone at a concert or event, where you might not meet again. In that case I think you would want to offer your contact info, facebook or whatever you are most comfortable sharing with someone you've just met.
If it is something inbetween those two, like you both shop at the same store or get coffee at the same place every day and just happen to run into each other, I would give it at least five times and start small, like saying Hi, Thank you, commenting on the weather and then work up to offering your info.
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Your Aspie score: 165 of 200
Your neurotypical (non-autistic) score: 48 of 200
EQ 12 SQ 70 = Extreme Systemizer