A003035 Maximal number of 3-tree rows in n-tree orchard problem.

0, 0, 1, 1, 2, 4, 6, 7, 10, 12, 16, 19, 22, 26

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• hard
• more

Comments

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

References

• P. Brass et al., Research Problems in Discrete Geometry, Springer, 2005.
• S. A. Burr, in The Mathematical Gardner, Ed. D. A. Klarner, p. 94, Wadsworth, 1981.
• S. A. Burr, B. Grünbaum and N. J. A. Sloane, The Orchard Problem, Geometriae Dedicata, 2 (1974), 397-424.
• Jean-Paul Delahaye, Des points qui s'alignent... ou pas, "Logique et calcul" column, "Pour la science", June 2021.
• H. E. Dudeney, Amusements in Mathematics, Nelson, London, 1917, page 56.
• 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.
• M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, Chap. 22.
• B. Grünbaum, Arrangements and Spreads. American Mathematical Society, Providence, RI, 1972, p. 22.
• John Jackson, Rational Amusements for Winter Evenings, London, 1821.
• F. Levi, Geometrische Konfigurationen, Hirzel, Leipzig, 1929.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• S. A. Burr, B. Grünbaum and N. J. A. Sloane, The Orchard Problem, Geometriae Dedicata, 2 (1974), 397-424.
• Zhao Hui Du, Orchard Planting Problem [From _Zhao Hui Du_, Nov 20 2008] [Seems to concentrate on the 4 trees per line version. - _N. J. A. Sloane_, Oct 16 2010]
• Noam D. Elkies, On some points-and-lines problems and configurations, arXiv:math/0612749 [math.MG], 2006; [Concerned with other versions of the problem].
• Erich Friedman, Table of values and bounds for up to 25 trees
• Z. Füredi and I. Palasti, Arrangements of lines with a large number of triangles, Proc. Amer. Math. Soc., 92(4):561-566, 1984.
• B. Green, T. Tao, On sets defining few ordinary lines, arXiv:1208.4714. (Shows that a(n) = [n(n-3)/6]+1 for all sufficiently large n.)
• R. Padmanabhan, Alok Shukla, Orchards in elliptic curves over finite fields, arXiv:2003.07172 [math.NT], 2020.
• Ed Pegg, Jr., Cultivating New Solutions for the Orchard-Planting Problem
• Ed Pegg, Jr., Illustration showing that a(15) >= 31 [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]}}]
• Ed Pegg, Jr., Illustration showing that a(15) >= 31 and a(16) >= 37
• Ed Pegg, Jr., Illustration for a(16) = 37 [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.]
• Ed Pegg, Jr., Illustrations of constructions for 9 through 28 trees.
• G. B. Purdy and J. W. Smith, Lines, circles, planes and spheres, Discrete Comput. Geom., 44 (2010), 860-882. [Makes use of A003035 in a formula. - _N. J. A. Sloane_, Oct 19 2017]
• N. J. A. Sloane, Illustration of initial terms (from Grünbaum-Burr-Sloane paper)
• J. Solymosi and M. Stojakovic, Many collinear k-tuples with no k + 1 collinear points, Discrete & Computational Geometry, October 2013, Volume 50, Issue 3, pp 811-820; also arXiv 1107.0327, 2013.
• Eric Weisstein's World of Mathematics, Orchard-Planting Problem.

Crossrefs

Cf. A006065 (4 trees/row), A008997 (5 trees per row), A058212.

Extensions

13 and 14 trees result from Zhao Hui Du, Nov 20 2008

Replaced my old picture with link to my write-up. - Ed Pegg Jr, Feb 02 2018