A248866 Discrete Heilbronn Triangle Problem: a(n) is twice the maximal area of the smallest triangle defined by three vertices that are a subset of n points on an n X n square lattice.

4, 9, 6, 6, 5, 6, 5, 6, 6, 6, 6

Offset

3,1

Comments

For n points in an n X n square, find the three points that make the triangle with minimal area. a(n) is double the maximal area of this triangle.

It is conjectured that the sequence has an infinite repetition of only two integers.

Links

Table of n, a(n) for n=3..13.

Gordon Hamilton, Unsolved K-12: Grade 8 Problems

Hiroaki Yamanouchi, examples for a(3)-a(13)

Example

a(3) = 4 because 3 points can be chosen so the minimal triangle has area 2:
.x.
...
x.x
a(6) = 6 because 3 points can be chosen so the minimal triangle has area 3:
..x..x
......
x.....
.....x
......
x..x..
a(8) is greater than or equal to 4 because of this non-optimal arrangement:
.....x.x
........
x.x.....
........
........
x.x.....
........
.....x.x
a(8) = 6 because 3 points can be chosen so the minimal triangle has area 3:
..x..x..
........
x......x
........
........
x......x
........
..x..x..

Crossrefs

Sequence in context: A108533 A200414 A347215 * A168608 A200383 A199513

Adjacent sequences: A248863 A248864 A248865 * A248867 A248868 A248869

Keyword

nonn,more

Author

Gordon Hamilton, Mar 04 2015

Extensions

a(5), a(7) and a(9) corrected and a(10)-a(13) added by Hiroaki Yamanouchi, Mar 09 2015

Status

approved