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This is a variant of the orchard-planting problem that uses circles instead of straight lines.
The maximal number of 3-point circles passing through n points on a plane is binomial(n,3). Given an arrangement of n points in general position, any choice of three points defines a circle. - Peter Kagey, Oct 05 2020
Paul Panzer provides upper and lower bounds:
a(n) <= floor[n*(n-1)*(n-2)/24].
a(n) >= 2 + n*[(n-2)*(n-2)+4]/32 for n == 0 mod 4 and n >= 8.
a(n) >= 2 + (n-1)*[(n-1)*(n-5)+16]/32 for n == 1 mod 4 and n >= 9.
a(n) >= 2 + n*(n-2)*(n-2)/32 for n == 2 mod 4 and n >= 10.
a(n) >= 2 + (n-1)*[(n-3)*(n-3)+16]/32 for n == 3 mod 4 and n >= 11.
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