%I A006066 M1334
%S A006066 0,0,1,2,5,7,11,15,21
%N A006066 Kobon triangles: maximal number of nonoverlapping triangles that can 
               be formed from n lines drawn in the plane.
%C A006066 The known values a = a(n) and upperbounds U (usually A032765(n)) with 
               name of discoverer of the arrangement when known are as follows:
%C A006066 n a U [Found by]
%C A006066 ---------------
%C A006066 1 0 0
%C A006066 2 0 0
%C A006066 3 1 1
%C A006066 4 2 2
%C A006066 5 5 5
%C A006066 6 7 7
%C A006066 7 11 11
%C A006066 8 15 16
%C A006066 9 21 21
%C A006066 10 25? 26 [Gruenbaum]
%C A006066 11 32? 33 [See link below]
%C A006066 12 ? 40
%C A006066 13 47 47 [Kabanovitch]
%C A006066 14 >= 53 56 [Bader]
%C A006066 15 65 65 [Suzuki]
%C A006066 16 >=72 74 [Bader]
%C A006066 17 85 85 [Bader]
%C A006066 18 >= 93 96 [Bader]
%C A006066 19 >= 104 107 [Bader]
%C A006066 20 >= 115 120 [Bader]
%C A006066 21 >= 130 133 [Bader]
%C A006066 22 ? 146
%C A006066 23 ? 161
%C A006066 24 ? 176
%C A006066 25 ? 191
%C A006066 26 ? 208
%C A006066 27 ? 225
%C A006066 28 ? 242
%C A006066 29 ? 261
%C A006066 30 ? 280
%C A006066 31 ? 299
%C A006066 32 ? 320
%C A006066 Ed Pegg's web page gives the upper bound for a(6) as 8. But by considering 
               all possible arrangements of 6 lines - the sixth term of A048872 
               - one can see that 8 is impossible. - N. J. A. Sloane (njas(AT)research.att.com), 
               Nov 11 2007
%C A006066 Although they are somewhat similar, this sequence is strictly different 
               from A084935, since A084935(12) = 48 exceeds the upper bound on a(12) 
               from A032765. - Floor en Lyanne van Lamoen (fvanlamoen(AT)planet.nl), 
               Nov 16 2005
%C A006066 The name is sometimes incorrectly entered as "Kodon" triangles.
%D A006066 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006066 M. Gardner, Wheels, Life and Other Mathematical Amusements. Freeman, 
               NY, 1983, p. 171.
%D A006066 Branko Gruenbaum, Convex Polytopes; p. 400 shows that a(10) >= 25.
%D A006066 Viatcheslav Kabanovitch, Kobon Triangle Solutions, Sharada (Charade, 
               by the Russian puzzle club Diogen), pp. 1-2, June 1999.
%H A006066 J. Bader, <a href="http://www.tik.ee.ethz.ch/sop/people/baderj/?page=other.php">
               Kobon Triangles</a>
%H A006066 J. Bader, <a href="http://www.tik.ee.ethz.ch/sop/people/baderj/Maximal17LinesKobonTriangleLarge.gif">
               Illustration showing a(17)=85</a>, Nov 28 2007.
%H A006066 S. Honma, <a href="http://www004.upp.so-net.ne.jp/s_honma/triangle/triangle2.htm">
               Title? (A related site)</a>
%H A006066 S. Honma, <a href="http://www10.plala.or.jp/rascalhp/image/10-25-2.gif">
               Title? (A related site)</a>
%H A006066 S. Honma, <a href="http://www10.plala.or.jp/rascalhp/image/11-32.gif">
               Illustration showing a(11)>=32</a>
%H A006066 S. Honma, <a href="http://www10.plala.or.jp/rascalhp/nlines.htm">Title? 
               (A related site)</a>
%H A006066 S. Honma, <a href="http://www10.plala.or.jp/rascalhp/nlines2.htm">Title? 
               (A related site)</a>
%H A006066 S. Honma, <a href="http://www10.plala.or.jp/rascalhp/nlines3.htm">Title? 
               (A related site)</a>
%H A006066 Ed Pegg, Jr., <a href="http://www.maa.org/editorial/mathgames/mathgames_02_08_06.html">
               Kobon triangles</a>
%H A006066 Alexandre Wajnberg, <a href="a006066.tiff">Illustration showing a(10) 
               >= 25</a> [A different construction from Gruenbaum's]
%H A006066 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               KobonTriangle.html">Kobon Triangle</a>
%F A006066 An upper bound on this sequence is given by A032765.
%e A006066 a(17) = 85 because the a configuration with 85 exists meeting the upper 
               bound.
%Y A006066 Sequence in context: A157001 A134640 A032616 this_sequence A084935 A062409 
               A089781
%Y A006066 Adjacent sequences: A006063 A006064 A006065 this_sequence A006067 A006068 
               A006069
%K A006066 nonn
%O A006066 1,4
%A A006066 N. J. A. Sloane (njas(AT)research.att.com).
%E A006066 a(15) = 65 found by T. Suzuki on Oct 02, 2005. - Eric Weisstein (eric(AT)weisstein.com), 
               Oct 04, 2005.
%E A006066 Gruenbaum reference from Anthony Labarre, Dec 19 2005
%E A006066 Additional links to Japanese web sites from Alexandre Wajnberg (alexandre.wajnberg(AT)skynet.be), 
               Dec 29 2005 and Anthony Labarre (alabarre(AT)ulb.ac.be), Dec 30 2005
%E A006066 A perfect solution for 13 lines was found in 1999 by Kabanovitch. - Ed 
               Pegg, Jr., Feb 08 2006
%E A006066 Updated with results from Johannes Bader (johannes.bader(AT)tik.ee.ethz.ch), 
               Dec 06 2007, who says "Acknowledgments and dedication to Corinne 
               Thomet".