A006065 Maximal number of 4-tree rows in n-tree orchard problem. (Formerly M0290)

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

Offset

1,7

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

Table of n, a(n) for n=1..20.

P. Berloquin, a(12) >= 7 (from an article in Jeux & Strategies from 1983 - see Fig. 10).

S. A. Burr, B. Grünbaum and N. J. A. Sloane, The Orchard Problem, Geometriae Dedicata, 2 (1974), 397-424.

S. A. Burr, B. 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)

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 [math.CO], 2011-2013.

Eric Weisstein's World of Mathematics, Orchard-Planting Problem.

Formula

a(n) >= A172992(n).

Crossrefs

Cf. A003035, A008997.

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

Sequence in context: A214045 A136432 A172992 * A218933 A266746 A096981

Adjacent sequences: A006062 A006063 A006064 * A006066 A006067 A006068

Keyword

nonn,hard,nice,more

Author

N. J. A. Sloane

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

Status

approved