A003035 Maximal number of 3-tree rows in n-tree orchard problem. (Formerly M0982)

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

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), 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

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), 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. Furedi 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.

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 write-up. - Ed Pegg Jr, Feb 02 2018

Status

approved