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A337747 Maximal number of 4-point circles passing through n points on a plane.

%I A337747 #77 Sep 05 2023 01:55:14
%S A337747 0,0,0,1,1,3,6,12,14,22,30,45
%N A337747 Maximal number of 4-point circles passing through n points on a plane.
%C A337747 This is a variant of the orchard-planting problem that uses circles instead of straight lines.
%C A337747 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
%C A337747 Paul Panzer provides upper and lower bounds:
%C A337747   a(n) <= floor(n*(n-1)*(n-2)/24).
%C A337747   a(n) >= 2 + n*((n-2)*(n-2) + 4)/32 for n == 0 (mod 4) and n >= 8.
%C A337747   a(n) >= 2 + (n-1)*((n-1)*(n-5) + 16)/32 for n == 1 (mod 4) and n >= 9.
%C A337747   a(n) >= 2 + n*(n-2)*(n-2)/32 for n == 2 (mod 4) and n >= 10.
%C A337747   a(n) >= 2 + (n-1)*((n-3)*(n-3) + 16)/32 for n == 3 (mod 4) and n >= 11.
%C A337747 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
%C A337747 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
%H A337747 Zhao Hui Du, <a href="https://github.com/emathgroup/selectedTopics/blob/master/content/attached/files/sd11tree2_12c.tgz">Source code and running data to show there are at most 30 circles for 11 trees</a>.
%H A337747 Zhao Hui Du, <a href="https://bbs.emath.ac.cn/forum.php?mod=redirect&amp;goto=findpost&amp;ptid=18664&amp;pid=93668">A group of solutions with 30 circles for 11 trees</a>.
%H A337747 Zhao Hui Du, <a href="https://github.com/emathgroup/selectedTopics/blob/master/content/attached/files/c12.ggb">Geogebra dynamic graph for a group of solution to 12 trees 45 circles (4 trees per circle)</a>.
%H A337747 Zhao Hui Du, <a href="/A337747/a337747_1.txt">Best known solution for 14 trees (73 circles)</a>, removing point I from the solution to form a solution with 13 trees and 53 circles so that a(13) >= 53, a(14) >= 73.
%H A337747 Zhao Hui Du, <a href="/A337747/a337747_1.png">Graph for 14 trees 73 circles</a> (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).
%H A337747 Zhao Hui Du, <a href="/A337747/a337747.jpg">Graph for 16 trees with 120 circles</a>.
%H A337747 Zhao Hui Du, <a href="https://github.com/emathgroup/selectedTopics/blob/master/content/attached/files/c16.html">Geogebra graph for 16 trees with 120 circles (parameter d and m could be any real value and point PP could be moving too), removing any tree to form 15 trees with 90 circles so that a[15]>=90, a[16]>=120</a>.
%H A337747 Zhao Hui Du, <a href="https://bbs.emath.ac.cn/forum.php?mod=redirect&amp;goto=findpost&amp;ptid=18664&amp;pid=93902">Chinese webpage to show that the number of 4 points circle through the n points in ellipse could be transformed to the number to pick 4 different numbers k1,k2,k3,k4 from 1 to n so that n|2+k1+k2+k3+k4, which is equal to A008610(n-4)</a>.
%H A337747 Dmitry Kamenetsky, <a href="/A337747/a337747.txt">Best known solutions for n <= 13</a>.
%H A337747 Dmitry Kamenetsky, <a href="https://puzzling.stackexchange.com/questions/101719/orchard-planting-problem-for-circles/">Orchard planting problem</a>, Puzzling StackExchange, August 2020.
%H A337747 Dmitry Kamenetsky, <a href="https://puzzling.stackexchange.com/questions/101945/general-orchard-planting-problem-for-circles/">General orchard planting problem</a>, Puzzling StackExchange, September 2020.
%e A337747 See examples in links.
%Y A337747 Cf. A003035 (the original orchard problem), A006065.
%K A337747 nonn,more,nice
%O A337747 1,6
%A A337747 _Dmitry Kamenetsky_, Sep 17 2020
%E A337747 a(11) from _Zhao Hui Du_, Nov 22 2022
%E A337747 a(12) from _Zhao Hui Du_, Dec 01 2022