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A003035 Maximal number of 3-tree rows in n-tree orchard problem.
(Formerly M0982)
4
0, 0, 1, 1, 2, 4, 6, 7, 10, 12, 16, 19, 22, 26 (list; graph; refs; listen; history; text; internal format)
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.

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.

F. Levi, Geometrische Konfigurationen, Hirzel, Leipzig, 1929.

Purdy, George B., and Justin W. Smith. "Lines, circles, planes and spheres." Discrete & Computational Geometry 44.4 (2010): 860-882. [Makes use of A003035 in a formula. - N. J. A. Sloane, Oct 19 2017]

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.

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].

Erich Friedman, Table of values and bounds for up to 25 trees

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.)

Ed Pegg, Jr., Illustration showing that a(15) >= 31.

Ed Pegg, Jr., Illustration showing that a(15) >= 31 [Another version that uses all 31 triplets 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.

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

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 Du, Zhao Hui, Nov 20 2008

STATUS

approved

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Last modified December 14 11:57 EST 2017. Contains 295981 sequences.