

A003034


Sylvester's problem: minimal number of ordinary lines through n points in the plane.
(Formerly M2275)


0



3, 3, 4, 3, 3, 4, 6, 5, 6, 6, 6, 7
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OFFSET

3,1


COMMENTS

An ordinary line contains exactly 2 points. The problem is to place n points, not all on a line, so as to minimize the number of ordinary lines.
Pach and Sharir give the following table for n >= 3: 3, 3, 4, 3, 3, 4, 6, 5, 6, 6, 6, 7, ?, 8, ?, 9, ?, ?, ?, 22, ...  N. J. A. Sloane, Dec 01 2010


REFERENCES

J. Brakke, Some new values for Sylvester's function for n collinear points, J. Undergrad. Math., 4 (1972), 1114.
H. T. Croft, K. J. Falconer and R. K. Guy, Unsolved Problems in Geometry, F12.
B. Grünbaum, Arrangements and Spreads. American Mathematical Society, Providence, RI, 1972, p. 18.
S. Hansen, Contributions to the SylvesterGallai theory, Dissertation, Univ. Copenhagen, 1981.
J. Pach and M. Sharir, Combinatorial Geometry and Its Algorithmic Applications, Amer. Math. Soc., 2009; see p. 2.
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=3..14.
P. Borwein and W. O. J. Moser, A survey of Sylvester's problem and its generalizations, Aequat. Math., 40 (1990), 111135.
D. W. Crowe and T. A. McKee, Sylvester's problem on collinear points, Math. Mag., 41 (1968), 3034.
L. M. Kelley and W. O. J. Moser, On the number of ordinary lines determined by n points, Canad. J. Math., 10 (1958), 210219.


FORMULA

Kelly and Moser showed that a(n) >= ceiling(3n/7); Hansen showed that a(n) >= floor(n/2) except for n=7 and 13.


CROSSREFS

Sequence in context: A188590 A080038 A121937 * A091282 A202708 A027684
Adjacent sequences: A003031 A003032 A003033 * A003035 A003036 A003037


KEYWORD

nonn,hard,nice


AUTHOR

N. J. A. Sloane


STATUS

approved



