This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A006066 M1334 #158 Aug 23 2025 09:30:44 %S A006066 0,0,1,2,5,7,11,15,21,25,32,38,47 %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 upper bounds 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 25 [Grünbaum] %C A006066 11 32 33 [32-triangle solutions found by Honma and Kabanovitch; proved maximal by Savchuk 2025] %C A006066 12 38 38 [Kabanovitch] %C A006066 13 47 47 [Kabanovitch] %C A006066 14 >= 53 54 [Bader] %C A006066 15 65 65 [Suzuki] %C A006066 16 72 72 [Bader] %C A006066 17 85 85 [Bader] %C A006066 18 >= 93 94 [Bader] %C A006066 19 107 107 [Wood] %C A006066 20 >= 116 117 [Wood] %C A006066 21 133 133 [Savchuk] %C A006066 22 >= 143 144 [Savchuk] %C A006066 23 161 161 [Savchuk] %C A006066 24 >= 172 173 [Savchuk] %C A006066 25 191 191 [Bartholdi] %C A006066 26 ? 205 %C A006066 27 225 225 [Savchuk] %C A006066 28 ? 239 %C A006066 29 261 261 [Bartholdi] %C A006066 30 ? 276 %C A006066 31 299 299 [Wood] %C A006066 32 ? 316 %C A006066 33 341 341 [Bartholdi] %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_, 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 van Lamoen_, Nov 16 2005 %C A006066 The name is sometimes incorrectly entered as "Kodon" triangles. %C A006066 Named after the Japanese puzzle expert and mathematics teacher Kobon Fujimura (1903-1983). - _Amiram Eldar_, Jun 19 2021 %D A006066 Nicolas Bartholdi, Jérémy Blanc, and Sébastien Loisel, "On simple arrangements of lines and pseudo-lines in P^2 and R^2 with the maximum number of triangles", 2008, in Goodman, Jacob E.; Pach, János; Pollack, Richard (eds.), Surveys on Discrete and Computational Geometry: Proceedings of the 3rd AMS-IMS-SIAM Joint Summer Research Conference "Discrete and Computational Geometry—Twenty Years Later" held in Snowbird, UT, June 18-22, 2006, Contemporary Mathematics, vol. 453, Providence, Rhode Island: American Mathematical Society, pp. 105-116, doi:10.1090/conm/453/08797, ISBN 978-0-8218-4239-3, MR 2405679 %D A006066 Martin Gardner, Wheels, Life and Other Mathematical Amusements, Freeman, NY, 1983, pp. 170, 171, 178. Mentions that the problem was invented by Kobon Fujimura. %D A006066 Branko Grünbaum, Convex Polytopes, Wiley, NY, 1967; 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. %D A006066 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %H A006066 Johannes Bader, <a href="https://www.sop.tik.ee.ethz.ch/people/baderj/other.html">Kobon Triangles</a>. %H A006066 Johannes Bader, <a href="/A006066/a006066_1.pdf">Kobon Triangles</a>. [Cached copy, with permission, pdf format] %H A006066 Johannes Bader, <a href="https://www.sop.tik.ee.ethz.ch/people/baderj/Maximal17LinesKobonTriangleLarge.gif">Illustration showing a(17)=85</a>, Nov 28 2007. %H A006066 Johannes Bader, <a href="/A006066/a006066_6.gif">Illustration showing a(17)=85</a>, Nov 28 2007. [Cached copy, with permission] %H A006066 Nicolas Bartholdi, Jérémy Blanc, and Sébastien Loisel, <a href="https://arxiv.org/abs/0706.0723">On simple arrangements of lines and pseudo-lines in P^2 and R^2 with the maximum number of triangles</a>, arXiv:0706.0723 [math.CO], 2007. %H A006066 Nicolas Bartholdi, Jérémy Blanc, Sébastien Loisel, and Pavlo Savchuk, <a href="/A006066/a006066_4.svg">Illustration showing a(33) = 341</a>, 2008. %H A006066 Gilles Clement and Johannes Bader, <a href="https://www.sop.tik.ee.ethz.ch/publicationListFiles/cb2007a.pdf">Tighter Upper Bound for the Number of Kobon Triangles</a>, Unpublished, 2007. %H A006066 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 A006066 Martin Gardner, <a href="/A005130/a005130_1.pdf">Letter to N. J. A. Sloane</a>, Jun 20 1991. %H A006066 S. Honma, <a href="https://web.archive.org/web/20171111045123/https://www004.upp.so-net.ne.jp/s_honma/triangle/triangle2.htm">三角形の最大数</a> %H A006066 S. Honma, <a href="https://web.archive.org/web/20171111045101/http://www10.plala.or.jp/rascalhp/image/10-25-2.gif">Illustration showing a(10)>=25</a> %H A006066 S. Honma, <a href="https://web.archive.org/web/20171111045215/http://www10.plala.or.jp/rascalhp/image/11-32.gif">Illustration showing a(11)>=32</a> %H A006066 S. Honma, n本の直線でn(n-2)/3個の三角形が出来る条件についての考察: <a href="https://web.archive.org/web/20171111045216/http://www10.plala.or.jp/rascalhp/nlines.htm">part 1</a>, <a href="https://web.archive.org/web/20221209001801/http://www10.plala.or.jp/rascalhp/nlines2.htm">part 2</a>, <a href="https://web.archive.org/web/20221208231339/http://www10.plala.or.jp/rascalhp/nlines3.htm">part 3</a>. %H A006066 Ed Pegg, Jr., <a href="https://www.mathpuzzle.com/MAA/45-Kobon/mathgames_02_08_06.html">Kobon triangles</a>, 2006. %H A006066 Ed Pegg, Jr., <a href="/A006066/a006066_2.pdf">Kobon Triangles</a>, 2006. [Cached copy, with permission, pdf format] %H A006066 Luis Felipe Prieto-Martínez, <a href="https://arxiv.org/abs/2104.09324">A list of problems in Plane Geometry with simple statement that remain unsolved</a>, arXiv:2104.09324 [math.HO], 2021. %H A006066 Pavlo Savchuk, <a href="https://arxiv.org/abs/2507.07951">Constructing Optimal Kobon Triangle Arrangements via Table Encoding, SAT Solving, and Heuristic Straightening</a>, arXiv:2507.07951 [math.CO], 2025. See pp. 1, 17. %H A006066 Pavlo Savchuk, <a href="/A006066/a006066.svg">Illustration showing a(21) = 133</a> %H A006066 Pavlo Savchuk, <a href="/A006066/a006066_1.svg">Illustration showing a(23) = 161</a> %H A006066 Pavlo Savchuk, <a href="/A006066/a006066_2.svg">Corresponding pseudo-line arrangement for the optimal n=23 solution</a> %H A006066 Pavlo Savchuk, <a href="/A006066/a006066_5.svg">Illustration showing a(27) = 225</a> %H A006066 Pavlo Savchuk, <a href="/A006066/a006066_7.svg">Illustration showing a(22) >= 143</a> %H A006066 Pavlo Savchuk, <a href="/A006066/a006066_8.svg">Illustration showing a(24) >= 172</a> %H A006066 N. J. A. Sloane, <a href="/A006066/a006066_5.jpg">Illustration for a(5) = 5</a> (a pentagram) %H A006066 Alexandre Wajnberg, <a href="/A006066/a006066.tiff">Illustration showing a(10) >= 25</a>. [A different construction from Grünbaum's] %H A006066 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KobonTriangle.html">Kobon Triangle</a>. %H A006066 Kyle Wood, <a href="/A006066/a006066.png">Illustration showing a(19) = 107</a> %H A006066 Kyle Wood, <a href="/A006066/a006066_3.svg">Illustration showing a(20) >= 116</a> %H A006066 Kyle Wood, <a href="/A006066/a006066_6.svg">Illustration showing a(31) = 299</a> %F A006066 An upper bound on this sequence is given by A032765. %F A006066 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 %F A006066 The upper bound for even n can be improved: floor(n(n-7/3)/3), proven by Bartholdi et. al. %e A006066 a(17) = 85 because the a configuration with 85 exists meeting the upper bound. %K A006066 nonn,nice,hard,more,changed %O A006066 1,4 %A A006066 _N. J. A. Sloane_ %E A006066 a(15) = 65 found by _Toshitaka Suzuki_ on Oct 02 2005. - _Eric W. Weisstein_, Oct 04 2005 %E A006066 Grünbaum reference from _Anthony Labarre_, Dec 19 2005 %E A006066 Additional links to Japanese web sites from _Alexandre Wajnberg_, Dec 29 2005 and _Anthony Labarre_, 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". %E A006066 a(11)-a(13) from _Eric W. Weisstein_, Jul 26 2025