A recursive formula for an upper bound:
a(n+1) <= (a(n)^4 - a(n)^2)/8 + (a(n) - a(n)^3)/4 which is equivalent to
a(n+1) <= binomial(binomial(a(n),2),2) (proven).
The proof of the above formula comes from the fact that if there are o points on a graph, then at most (o^2-o)/2 lines that can be drawn between them. If there are m lines on a graph, then there are at most (m^2-m)/2 intersections between them; substituting and simplifying leads to the former upper limit.
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