A248866 - OEIS

%I #22 Mar 13 2015 15:14:59

%S 4,9,6,6,5,6,5,6,6,6,6

%N 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.

%C 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.

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

%H Gordon Hamilton, <a href="http://youtu.be/rz5Ap8YnWoo">Unsolved K-12: Grade 8 Problems</a>

%H Hiroaki Yamanouchi, <a href="/A248866/a248866.txt">examples for a(3)-a(13)</a>

%e a(3) = 4 because 3 points can be chosen so the minimal triangle has area 2:

%e .x.

%e ...

%e x.x

%e a(6) = 6 because 3 points can be chosen so the minimal triangle has area 3:

%e ..x..x

%e ......

%e x.....

%e .....x

%e ......

%e x..x..

%e a(8) is greater than or equal to 4 because of this non-optimal arrangement:

%e .....x.x

%e ........

%e x.x.....

%e ........

%e ........

%e x.x.....

%e ........

%e .....x.x

%e a(8) = 6 because 3 points can be chosen so the minimal triangle has area 3:

%e ..x..x..

%e ........

%e x......x

%e ........

%e ........

%e x......x

%e ........

%e ..x..x..

%K nonn,more

%O 3,1

%A _Gordon Hamilton_, Mar 04 2015

%E a(5), a(7) and a(9) corrected and a(10)-a(13) added by _Hiroaki Yamanouchi_, Mar 09 2015