A006065
Maximal number of 4-tree rows in n-tree orchard problem.
Original entry on oeis.org
0, 0, 0, 1, 1, 1, 2, 2, 3, 5, 6, 7, 9, 10, 12, 15, 16, 18, 20, 23
Offset: 1
- M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, Chap. 22.
- F. Levi, Geometrische Konfigurationen, Hirzel, Leipzig, 1929.
- Xianzu Lin, A new result about orchard-planting problem, Preprint, 2005. [Shows a(20) >= 23.]
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- For further references and links see A003035.
- P. Berloquin, a(12) >= 7 (from an article in Jeux & Strategies from 1983 - see Fig. 10).
- Thomas Bloom, Problem 669 and possibly Problem 101, Erdős Problems.
- Stefan A. Burr, Branko Grünbaum, and N. J. A. Sloane, The Orchard Problem, Geometriae Dedicata, 2 (1974), 397-424.
- Stefan A. Burr, Branko Grünbaum, and N. J. A. Sloane, The Orchard Problem, Geometriae Dedicata, 2 (1974), 397-424.
- Zhao Hui Du, Code to verify a(13) to a(16) for orchard planting problem
- Zhao Hui Du, Full list of the optimal results from 13~18 trees
- Zhao Hui Du, A Chinese webpage for the problem
- Zhao Hui Du, Illustration showing that a(22)>=28 [Line ABCV is infinity line]
- Noam D. Elkies, On some points-and-lines problems and configurations, arXiv:math/0612749 [math.MG], 2006.
- Erich Friedman, Table of values and bounds for up to 25 trees
- Branko Grünbaum and J. F. Rigby, The real configuration (21_4), Journal of the London Mathematical Society 2.2 (1990): 336-346. [Shows a(21) >= 21.]
- Xianzu Lin, Illustration showing that a(20) >= 23 [The points S and T are at infinity]
- Ed Pegg, Jr., Cultivating New Solutions for theOrchard-Planting Problem, 2018.
- Ed Pegg, Jr., Mathpuzzxle Blog, Updated Feb 27 2020. [Gives new construction for n = 22]
- Ed Pegg, Jr., Mathpuzzxle Blog, Updated Feb 27 2020. [Gives new construction for n = 22] (extract, local copy)
- József Solymosi and Miloš Stojaković, 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 [math.CO], 2011-2013.
- Terence Tao, Erdős problem database, see nos. 101, 669.
- Eric Weisstein's World of Mathematics, Orchard-Planting Problem.
Cf.
A172992 (the same problem, but with integer-valued tree coordinates).
A003035
Maximal number of 3-tree rows in n-tree orchard problem.
Original entry on oeis.org
0, 0, 1, 1, 2, 4, 6, 7, 10, 12, 16, 19, 22, 26
Offset: 1
- 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).
- 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.
Replaced my old picture with link to my write-up. -
Ed Pegg Jr, Feb 02 2018
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