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 A006066 Kobon triangles: maximal number of nonoverlapping triangles that can be formed from n lines drawn in the plane. (Formerly M1334) 2

%I M1334

%S 0,0,1,2,5,7,11,15,21

%N Kobon triangles: maximal number of nonoverlapping triangles that can be formed from n lines drawn in the plane.

%C The known values a = a(n) and upper bounds U (usually A032765(n)) with name of discoverer of the arrangement when known are as follows:

%C n a U [Found by]

%C ---------------

%C 1 0 0

%C 2 0 0

%C 3 1 1

%C 4 2 2

%C 5 5 5

%C 6 7 7

%C 7 11 11

%C 8 15 16

%C 9 21 21

%C 10 25? 26 [Grünbaum]

%C 11 32? 33 [See link below]

%C 12 ? 40

%C 13 47 47 [Kabanovitch]

%C 14 >= 53 56 [Bader]

%C 15 65 65 [Suzuki]

%C 18 >= 93 96 [Bader]

%C 19 >= 104 107 [Bader]

%C 20 >= 115 120 [Bader]

%C 21 >= 130 133 [Bader]

%C 22 ? 146

%C 23 ? 161

%C 24 ? 176

%C 25 ? 191

%C 26 ? 208

%C 27 ? 225

%C 28 ? 242

%C 29 ? 261

%C 30 ? 280

%C 31 ? 299

%C 32 ? 320

%C 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_, Nov 11 2007

%C 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 van Lamoen_, Nov 16 2005

%C The name is sometimes incorrectly entered as "Kodon" triangles.

%D M. Gardner, Wheels, Life and Other Mathematical Amusements. Freeman, NY, 1983, pp. 170, 171, 178. Mentions that the problem was invented by Kobon Fujimura.

%D Branko Grünbaum, Convex Polytopes, Wiley, NY, 1967; p. 400 shows that a(10) >= 25.

%D Viatcheslav Kabanovitch, Kobon Triangle Solutions, Sharada (Charade, by the Russian puzzle club Diogen), pp. 1-2, June 1999.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H J. Bader, <a href="/A006066/a006066_1.pdf">Kobon Triangles</a> [Cached copy, with permission, pdf format]

%H J. Bader, <a href="/A006066/a006066_6.gif">Illustration showing a(17)=85</a>, Nov 28 2007. [Cached copy, with permission]

%H Gilles Clement and Johannes Bader, <a href="http://www.tik.ee.ethz.ch/sop/publicationListFiles/cb2007a.pdf">Tighter Upper Bound for the Number of Kobon Triangles</a>, Unpublished, 2007

%H Gilles Clement and Johannes Bader, <a href="/A006066/a006066.pdf">Tighter Upper Bound for the Number of Kobon Triangles</a>, Unpublished, 2007 [Cached copy, with permission]

%H M. Gardner, <a href="/A005130/a005130_1.pdf">Letter to N. J. A. Sloane</a>, Jun 20 1991.

%H S. Honma, <a href="http://www004.upp.so-net.ne.jp/s_honma/triangle/triangle2.htm">Title? (A related site)</a>

%H S. Honma, <a href="http://www10.plala.or.jp/rascalhp/image/10-25-2.gif">Title? (A related site)</a>

%H S. Honma, <a href="http://www10.plala.or.jp/rascalhp/image/11-32.gif">Illustration showing a(11)>=32</a>

%H S. Honma, <a href="http://www10.plala.or.jp/rascalhp/nlines.htm">Title? (A related site)</a>

%H S. Honma, <a href="http://www10.plala.or.jp/rascalhp/nlines2.htm">Title? (A related site)</a>

%H S. Honma, <a href="http://www10.plala.or.jp/rascalhp/nlines3.htm">Title? (A related site)</a>

%H Ed Pegg, Jr., <a href="http://www.maa.org/editorial/mathgames/mathgames_02_08_06.html">Kobon triangles</a>

%H Ed Pegg, Jr., <a href="/A006066/a006066_2.pdf">Kobon Triangles</a> [Cached copy, with permission, pdf format]

%H N. J. A. Sloane, <a href="/A006066/a006066_5.jpg">Illustration for a(5) = 5</a> (a pentagram)

%H Alexandre Wajnberg, <a href="/A006066/a006066.tiff">Illustration showing a(10) >= 25</a> [A different construction from Grünbaum's]

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KobonTriangle.html">Kobon Triangle</a>

%F An upper bound on this sequence is given by A032765.

%F For any odd n > 1, if n == 1 (mod 6), a(n) <= (n^2 - (2n + 2))/3; in other odd cases, a(n) <= (n^2 - 2n)/3. For any even n > 0, if n == 4 (mod 6), a(n) <= (n^2 - (2n + 2))/3, otherwise a(n) <= (n^2 - 2n)/3. - _Sergey Pavlov_, Feb 11 2017

%e a(17) = 85 because the a configuration with 85 exists meeting the upper bound.

%K nonn,hard,more

%O 1,4

%A _N. J. A. Sloane_

%E a(15) = 65 found by _Toshitaka Suzuki_ on Oct 02 2005. - _Eric W. Weisstein_, Oct 04 2005

%E Grünbaum reference from _Anthony Labarre_, Dec 19 2005

%E Additional links to Japanese web sites from _Alexandre Wajnberg_, Dec 29 2005 and _Anthony Labarre_, Dec 30 2005

%E A perfect solution for 13 lines was found in 1999 by Kabanovitch. - _Ed Pegg Jr_, Feb 08 2006

%E Updated with results from Johannes Bader (johannes.bader(AT)tik.ee.ethz.ch), Dec 06 2007, who says "Acknowledgments and dedication to Corinne Thomet".

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Last modified January 22 07:33 EST 2019. Contains 319353 sequences. (Running on oeis4.)