

A003035


Maximal number of 3tree rows in ntree orchard problem.
(Formerly M0982)


4



0, 0, 1, 1, 2, 4, 6, 7, 10, 12, 16, 19, 22, 26
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graph;
refs;
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OFFSET

1,5


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), 397424.
H. E. Dudeney, Amusements in Mathematics, Nelson, London, 1917, page 56.
Paul Erdos and George Purdy. Extremal problems in geometry, Chapter 17, pages 809874 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

Table of n, a(n) for n=1..14.
S. A. Burr, B. Grünbaum and N. J. A. Sloane, The Orchard Problem, Geometriae Dedicata, 2 (1974), 397424.
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 pointsandlines 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. Furedi and I. Palasti, Arrangements of lines with a large number of triangles, Proc. Amer. Math. Soc., 92(4):561566, 1984.
B. Green, T. Tao, On sets defining few ordinary lines, arXiv:1208.4714. (Shows that a(n) = [n(n3)/6]+1 for all sufficiently large n.)
Ed Pegg, Jr., Cultivating New Solutions for the OrchardPlanting 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 BurrGrünbaumSloane (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), 860882. [Makes use of A003035 in a formula.  N. J. A. Sloane, Oct 19 2017]
N. J. A. Sloane, Illustration of initial terms (from GrünbaumBurrSloane paper)
J. Solymosi and M. Stojakovic, Many collinear ktuples with no k + 1 collinear points, Discrete & Computational Geometry, October 2013, Volume 50, Issue 3, pp 811820; also arXiv 1107.0327, 2013.
Eric Weisstein's World of Mathematics, OrchardPlanting Problem.


CROSSREFS

Cf. A006065 (4 trees/row), A008997 (5 trees per row), A058212.
Sequence in context: A198033 A071260 A026407 * A094453 A191200 A026398
Adjacent sequences: A003032 A003033 A003034 * A003036 A003037 A003038


KEYWORD

nonn,nice,hard,more


AUTHOR

N. J. A. Sloane.


EXTENSIONS

13 and 14 trees result from Zhao Hui Du, Nov 20 2008
Replaced my old picture with link to my writeup.  Ed Pegg Jr, Feb 02 2018


STATUS

approved



