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%I M0982 #115 Aug 15 2024 10:58:33
%S 0,0,1,1,2,4,6,7,10,12,16,19,22,26
%N Maximal number of 3-tree rows in n-tree orchard problem.
%C It is known that a(15) is 31 or 32, a(16)=37 and a(17) is 40, 41 or 42. - _N. J. A. Sloane_, Feb 11 2013
%D P. Brass et al., Research Problems in Discrete Geometry, Springer, 2005.
%D S. A. Burr, in The Mathematical Gardner, Ed. D. A. Klarner, p. 94, Wadsworth, 1981.
%D S. A. Burr, B. Grünbaum and N. J. A. Sloane, The Orchard Problem, Geometriae Dedicata, 2 (1974), 397-424.
%D Jean-Paul Delahaye, Des points qui s'alignent... ou pas, "Logique et calcul" column, "Pour la science", June 2021.
%D H. E. Dudeney, Amusements in Mathematics, Nelson, London, 1917, page 56.
%D Paul Erdos and George Purdy. Extremal problems in geometry, Chapter 17, pages 809-874 in R. L. Graham et al., eds., Handbook of Combinatorics, 2 vols., MIT Press, 1995. See Section 3.7.
%D M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, Chap. 22.
%D B. Grünbaum, Arrangements and Spreads. American Mathematical Society, Providence, RI, 1972, p. 22.
%D John Jackson, Rational Amusements for Winter Evenings, London, 1821.
%D F. Levi, Geometrische Konfigurationen, Hirzel, Leipzig, 1929.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H S. A. Burr, B. Grünbaum and N. J. A. Sloane, <a href="http://neilsloane.com/doc/ORCHARD/orchard.html">The Orchard Problem</a>, Geometriae Dedicata, 2 (1974), 397-424.
%H Zhao Hui Du, <a href="http://zdu.spaces.live.com/blog/cns!C95152CB25EF2037!122.entry">Orchard Planting Problem</a> [From _Zhao Hui Du_, Nov 20 2008] [Seems to concentrate on the 4 trees per line version. - _N. J. A. Sloane_, Oct 16 2010]
%H Noam D. Elkies, <a href="https://arxiv.org/abs/math/0612749">On some points-and-lines problems and configurations</a>, arXiv:math/0612749 [math.MG], 2006; [Concerned with other versions of the problem].
%H Erich Friedman, <a href="https://erich-friedman.github.io/packing/trees/">Table of values and bounds for up to 25 trees</a>
%H Z. Füredi and I. Palasti, <a href="https://doi.org/10.1090/S0002-9939-1984-0760946-2">Arrangements of lines with a large number of triangles</a>, Proc. Amer. Math. Soc., 92(4):561-566, 1984.
%H B. Green, T. Tao, <a href="https://arxiv.org/abs/1208.4714">On sets defining few ordinary lines</a>, arXiv:1208.4714. (Shows that a(n) = [n(n-3)/6]+1 for all sufficiently large n.)
%H R. Padmanabhan, Alok Shukla, <a href="https://arxiv.org/abs/2003.07172">Orchards in elliptic curves over finite fields</a>, arXiv:2003.07172 [math.NT], 2020.
%H Ed Pegg, Jr., <a href="http://blog.wolfram.com/2018/02/02/cultivating-new-solutions-for-the-orchard-planting-problem/">Cultivating New Solutions for the Orchard-Planting Problem</a>
%H Ed Pegg, Jr., <a href="/A003035/a003035_1.jpg">Illustration showing that a(15) >= 31</a> [Another version that uses all 31 triples from -7 to 7 which sum to 0 (mod 15). Coordinates are: {-7, {-1 - Sqrt[3], -1 + 2 Sqrt[3]}}, {-6, {2 (2 + Sqrt[3]), -5}}, {-5, {0, -3}}, {-4, {-2 (2 + Sqrt[3]), -1}}, {-3, {-2, 1}}, {-2, {2, -1}}, {-1, {2 (2 + Sqrt[3]), 1}}, {0, {0, 3}}, {1, {-2 (2 + Sqrt[3]), 5}}, {2, {1 + Sqrt[3], 1 - 2 Sqrt[3]}}, {3, {-2 (2 + Sqrt[3]), -1 - 2 Sqrt[3]}}, {4, {-2 - Sqrt[3], 1}}, {5, {0, 0}}, {6, {2 + Sqrt[3], -1}}, {7, {2 (2 + Sqrt[3]), 1 + 2 Sqrt[3]}}]
%H Ed Pegg, Jr., <a href="/A003035/a003035_3.jpg">Illustration showing that a(15) >= 31 and a(16) >= 37</a>
%H Ed Pegg, Jr., <a href="/A003035/a003035_2.jpg">Illustration for a(16) = 37</a> [Based on a drawing in Burr-Grünbaum-Sloane (1974). The bottom left point is at -(sqrt(3), sqrt(5)). Note that 3 points and one line are at infinity.]
%H Ed Pegg, Jr., <a href="/A003035/a003035_4.jpg">Illustrations of constructions for 9 through 28 trees.</a>
%H G. B. Purdy and J. W. Smith, <a href="https://doi.org/10.1007/s00454-010-9270-3">Lines, circles, planes and spheres</a>, Discrete Comput. Geom., 44 (2010), 860-882. [Makes use of A003035 in a formula. - _N. J. A. Sloane_, Oct 19 2017]
%H N. J. A. Sloane, <a href="/A003035/a003035.gif">Illustration of initial terms (from Grünbaum-Burr-Sloane paper)</a>
%H J. Solymosi and M. Stojakovic, <a href="https://doi.org/10.1007/s00454-013-9526-9">Many collinear k-tuples with no k + 1 collinear points</a>, Discrete & Computational Geometry, October 2013, Volume 50, Issue 3, pp 811-820; also arXiv 1107.0327, 2013.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Orchard-PlantingProblem.html">Orchard-Planting Problem.</a>
%Y Cf. A006065 (4 trees/row), A008997 (5 trees per row), A058212.
%K nonn,nice,hard,more
%O 1,5
%A _N. J. A. Sloane_
%E 13 and 14 trees result from _Zhao Hui Du_, Nov 20 2008
%E Replaced my old picture with link to my write-up. - _Ed Pegg Jr_, Feb 02 2018
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