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%I #45 Mar 02 2021 19:13:06
%S 0,0,0,1,1,1,2,2,3,5,6,7,9,10,12,14,15,17,20,23,24,26,28,30,32,34,36,
%T 39,41,44,46,48,50,52,54,57,59,62,64,67,69,72,75,78,80,83,86,89,91,94,
%U 97,100,103,106,109,112,116,120,123,126
%N Maximal number of 4-tree rows in n-tree orchard problem in which the tree coordinates are integers.
%C From _Dmitry Kamenetsky_, Feb 23 2012: (Start)
%C At the end of 2011 there was a 3-month programming competition (http://infinitesearchspace.dyndns.org/orchardplanting) for this problem for n in the range [11,60]. Later there was an unofficial contest (http://infinitesearchspace.dyndns.org/content/orchard-mini-contest) for finding optimal solutions that can be placed on the smallest possible grids.
%C The terms in this sequences are the best known solutions to date and they may not be optimal.
%C Any solution that can be represented with integer coordinates can also be represented with real-valued coordinates, hence A172992(n)<=A006065(n). It is believed (but not proved) that some real-valued solutions cannot be represented by integer-valued solutions. For example A172992(16)=14, but A006065(16)=15. (End)
%H Zhao Hui Du, <a href="http://zdu.spaces.live.com/blog/cns!C95152CB25EF2037!122.entry">More detailed results</a> (no longer valid)
%H Infinite Search Space, <a href="https://web.archive.org/web/20121031235129/http://infinitesearchspace.dyndns.org/orchardplanting/tops">Competition for the tree-planting problem</a>, 2011.
%H Infinite Search Space, <a href="http://web.archive.org/web/20120120174947/http://infinitesearchspace.dyndns.org/content/orchard-mini-contest">Orchard mini-contest</a>, 2012.
%H Dmitry Kamenetsky, <a href="/A172992/a172992_2.txt">Best known solutions for n <= 29</a>.
%H Hugo Pfoertner, <a href="/A172992/a172992.png">Illustration of a(10) = 5</a>.
%e It is possible to place 10 points on an integer grid such that there are 5 straight lines with exactly 4 points lying on each line. Hence a(10) is 5. - _Dmitry Kamenetsky_, Feb 23 2012 (see the illustration by _Hugo Pfoertner_!)
%Y A006065 is the same problem, but with real-valued tree coordinates.
%K hard,nonn
%O 1,7
%A _Zhao Hui Du_, Feb 07 2010
%E More terms added by _Dmitry Kamenetsky_, Feb 23 2012
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