Based on my experience the couple times I stayed awake for more than 24 hours in a row, I think it's very unlikely to be quadratic, at least for large N.
So you are saying you did spent something like 36 hours awake in a row without negative side effects?
Yes, there were negative side effects, but these didn't include having to sleep 36^2/32 hours in a row to catch up.
(Edit: what happened is I slept six-ish hours in a row as soon as I hit a bed, waking up in the afternoon, then I reverted to my ordinary sleep schedule except the first couple nights I went to bed about an hour earlier than usual.
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