A006065 Maximal number of 4-tree rows in n-tree orchard problem.

0, 0, 0, 1, 1, 1, 2, 2, 3, 5, 6, 7, 9, 10, 12, 15, 16, 18, 20, 23

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• nice
• more
• changed

Comments

Maximum number of rows with exactly 4 trees in each row if there are n trees in the orchard.

For further references and links see A003035.

References

• 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.

Links

• 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.

Crossrefs

Cf. A003035, A008997.

Cf. A172992 (the same problem, but with integer-valued tree coordinates).

Formula

a(n) >= A172992(n).

Extensions

a(13)-a(15) from Zhao Hui Du, Aug 24 2008

a(17) from Zhao Hui Du, Nov 11 2008

a(18) from Zhao Hui Du, Nov 25 2008

a(19) from Zhao Hui Du, Dec 17 2009

a(20) from Zhao Hui Du, Feb 01 2010