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A337747
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Maximal number of 4-point circles passing through n points on a plane.
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1
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0, 0, 0, 1, 1, 3, 6, 12, 14, 22, 30, 45
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OFFSET
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1,6
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COMMENTS
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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.
It seems that a(n) = n*((n-2)*(n-2) + 4)/32 + 2*A008610(n/2-4) if n == 0 (mod 4) and n >= 8. - Zhao Hui Du, Dec 14 2022
The number of 4-point circles passing through n points (2*cos(t_k), sin(t_k)) where t_k = (2k-1)*Pi/n, k=1,2,...,n is A008610(n-4), so A337747(n) >= A008610(n-4), so A337747(n) ~ n^3/24 for sufficiently large n. - Zhao Hui Du, Dec 15 2022
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LINKS
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Zhao Hui Du, Graph for 14 trees 73 circles (all black circles have same patterns as that in that in two co-center regular polygons (Paul Panzer's solution) while red circles are those extra circles).
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EXAMPLE
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See examples in links.
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CROSSREFS
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KEYWORD
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nonn,more,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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