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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 (list; graph; refs; listen; history; text; internal format)



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


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.

H. E. Dudeney, Amusements in Mathematics, Nelson, London, 1917, page 56.

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.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


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.

Z. H. Du, Orchard Planting Problem [From Du, Zhao Hui, 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].

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.)

G. B. Purdy and J. W. Smith, Lines, circles, planes and spheres, Discrete Comput. Geom., 44 (2010), 860-882.

N. J. A. Sloane, Illustration of initial terms (from Grünbaum-Burr-Sloane paper)

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


Cf. A006065, A008997, A058212.

Sequence in context: A198033 A071260 A026407 * A094453 A191200 A026398

Adjacent sequences:  A003032 A003033 A003034 * A003036 A003037 A003038




N. J. A. Sloane.


13 and 14 trees result from Du, Zhao Hui, Nov 20 2008



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Last modified October 7 00:58 EDT 2015. Contains 262347 sequences.