cp's OEIS Frontend

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

%I A006065 M0290 #98 Sep 04 2025 10:54:21
%S A006065 0,0,0,1,1,1,2,2,3,5,6,7,9,10,12,15,16,18,20,23
%N A006065 Maximal number of 4-tree rows in n-tree orchard problem.
%C A006065 Maximum number of rows with exactly 4 trees in each row if there are n trees in the orchard.
%C A006065 For further references and links see A003035.
%D A006065 M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, Chap. 22.
%D A006065 F. Levi, Geometrische Konfigurationen, Hirzel, Leipzig, 1929.
%D A006065 Xianzu Lin, A new result about orchard-planting problem, Preprint, 2005. [Shows a(20) >= 23.]
%D A006065 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A006065 For further references and links see A003035.
%H A006065 P. Berloquin, <a href="/A008997/a008997.jpg">a(12) >= 7</a> (from an article in Jeux & Strategies from 1983 - see Fig. 10).
%H A006065 Thomas Bloom, <a href="https://www.erdosproblems.com/669">Problem 669</a> and possibly <a href="https://www.erdosproblems.com/101">Problem 101</a>, Erdős Problems.
%H A006065 Stefan A. Burr, Branko Grünbaum, and N. J. A. Sloane, <a href="https://doi.org/10.1007/BF00147569">The Orchard Problem</a>, Geometriae Dedicata, 2 (1974), 397-424.
%H A006065 Stefan A. Burr, Branko 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 A006065 Zhao Hui Du, <a href="http://bbs.emath.ac.cn/viewthread.php?tid=703&amp;page=11#pid9076">Code to verify a(13) to a(16) for orchard planting problem</a>
%H A006065 Zhao Hui Du, <a href="https://emathgroup.github.io/blog/orchard-planting-problem#fulllist">Full list of the optimal results from 13~18 trees</a>
%H A006065 Zhao Hui Du, <a href="http://bbs.emath.ac.cn/thread-703-1-1.html">A Chinese webpage for the problem</a>
%H A006065 Zhao Hui Du, <a href="/A006065/a006065.png">Illustration showing that a(22)>=28</a> [Line ABCV is infinity line]
%H A006065 Noam D. Elkies, <a href="http://arxiv.org/abs/math/0612749">On some points-and-lines problems and configurations</a>, arXiv:math/0612749 [math.MG], 2006.
%H A006065 Erich Friedman, <a href="https://erich-friedman.github.io/packing/trees/">Table of values and bounds for up to 25 trees</a>
%H A006065 Branko Grünbaum and J. F. Rigby, <a href="https://citeseerx.ist.psu.edu/pdf/7804e7b0fefa5dbd881a1a7102946b24098014d5">The real configuration (21_4)</a>, Journal of the London Mathematical Society 2.2 (1990): 336-346. [Shows a(21) >= 21.]
%H A006065 Xianzu Lin, <a href="/A006065/a006065.gif">Illustration showing that a(20) >= 23</a> [The points S and T are at infinity]
%H A006065 Ed Pegg, Jr., <a href="http://blog.wolfram.com/2018/02/02/cultivating-new-solutions-for-the-orchard-planting-problem/">Cultivating New Solutions for theOrchard-Planting Problem</a>, 2018.
%H A006065 Ed Pegg, Jr., <a href="http://www.mathpuzzle.com/">Mathpuzzxle Blog</a>, Updated Feb 27 2020. [Gives new construction for n = 22]
%H A006065 Ed Pegg, Jr., <a href="/A006065/a006065_1.png">Mathpuzzxle Blog</a>, Updated Feb 27 2020. [Gives new construction for n = 22] (extract, local copy)
%H A006065 József Solymosi and Miloš Stojaković, <a href="https://arxiv.org/abs/1107.0327">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 [math.CO], 2011-2013.
%H A006065 Terence Tao, <a href="https://github.com/teorth/erdosproblems/blob/main/README.md#table">Erdős problem database</a>, see nos. 101, 669.
%H A006065 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Orchard-PlantingProblem.html">Orchard-Planting Problem</a>.
%F A006065 a(n) >= A172992(n).
%Y A006065 Cf. A003035, A008997.
%Y A006065 Cf. A172992 (the same problem, but with integer-valued tree coordinates).
%K A006065 nonn,hard,nice,more,changed
%O A006065 1,7
%A A006065 _N. J. A. Sloane_
%E A006065 a(13)-a(15) from _Zhao Hui Du_, Aug 24 2008
%E A006065 a(17) from _Zhao Hui Du_, Nov 11 2008
%E A006065 a(18) from _Zhao Hui Du_, Nov 25 2008
%E A006065 a(19) from _Zhao Hui Du_, Dec 17 2009
%E A006065 a(20) from _Zhao Hui Du_, Feb 01 2010