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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
 0, 0, 1, 2, 5, 7, 11, 15, 21 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS 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: n a U [Found by] --------------- 1 0 0 2 0 0 3 1 1 4 2 2 5 5 5 6 7 7 7 11 11 8 15 16 9 21 21 10 25? 26 [Grünbaum] 11 32? 33 [See link below] 12 ? 40 13 47 47 [Kabanovitch] 14 >= 53 56 [Bader] 15 65 65 [Suzuki] 16 >=72 74 [Bader] 17 85 85 [Bader] 18 >= 93 96 [Bader] 19 >= 104 107 [Bader] 20 >= 115 120 [Bader] 21 >= 130 133 [Bader] 22 ? 146 23 ? 161 24 ? 176 25 ? 191 26 ? 208 27 ? 225 28 ? 242 29 ? 261 30 ? 280 31 ? 299 32 ? 320 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 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 The name is sometimes incorrectly entered as "Kodon" triangles. REFERENCES M. Gardner, Wheels, Life and Other Mathematical Amusements. Freeman, NY, 1983, pp. 170, 171, 178. Mentions that the problem was invented by Kobon Fujimura. Branko Grünbaum, Convex Polytopes, Wiley, NY, 1967; p. 400 shows that a(10) >= 25. Viatcheslav Kabanovitch, Kobon Triangle Solutions, Sharada (Charade, by the Russian puzzle club Diogen), pp. 1-2, June 1999. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS J. Bader, Kobon Triangles J. Bader, Kobon Triangles [Cached copy, with permission, pdf format] J. Bader, Illustration showing a(17)=85, Nov 28 2007. J. Bader, Illustration showing a(17)=85, Nov 28 2007. [Cached copy, with permission] Gilles Clement and Johannes Bader, Tighter Upper Bound for the Number of Kobon Triangles, Unpublished, 2007 Gilles Clement and Johannes Bader, Tighter Upper Bound for the Number of Kobon Triangles, Unpublished, 2007 [Cached copy, with permission] M. Gardner, Letter to N. J. A. Sloane, Jun 20 1991. S. Honma, Title? (A related site) S. Honma, Title? (A related site) S. Honma, Illustration showing a(11)>=32 S. Honma, Title? (A related site) S. Honma, Title? (A related site) S. Honma, Title? (A related site) Ed Pegg, Jr., Kobon triangles Ed Pegg, Jr., Kobon Triangles [Cached copy, with permission, pdf format] N. J. A. Sloane, Illustration for a(5) = 5 (a pentagram) Alexandre Wajnberg, Illustration showing a(10) >= 25 [A different construction from Grünbaum's] Eric Weisstein's World of Mathematics, Kobon Triangle FORMULA An upper bound on this sequence is given by A032765. 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 EXAMPLE a(17) = 85 because the a configuration with 85 exists meeting the upper bound. CROSSREFS Sequence in context: A216094 A184857 A032616 * A084935 A239072 A217302 Adjacent sequences:  A006063 A006064 A006065 * A006067 A006068 A006069 KEYWORD nonn,hard,more AUTHOR EXTENSIONS a(15) = 65 found by Toshitaka Suzuki on Oct 02 2005. - Eric W. Weisstein, Oct 04 2005 Grünbaum reference from Anthony Labarre, Dec 19 2005 Additional links to Japanese web sites from Alexandre Wajnberg, Dec 29 2005 and Anthony Labarre, Dec 30 2005 A perfect solution for 13 lines was found in 1999 by Kabanovitch. - Ed Pegg Jr, Feb 08 2006 Updated with results from Johannes Bader (johannes.bader(AT)tik.ee.ethz.ch), Dec 06 2007, who says "Acknowledgments and dedication to Corinne Thomet". STATUS approved

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Last modified August 17 22:40 EDT 2017. Contains 290680 sequences.