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%I #33 Dec 21 2022 08:19:14
%S 3,4,4,4,4,4,4,4,5,5,4,5,5,6,6,4,5,5,6,6,6,4,5,6,6,6,7,7,4,5,7,6,7,7,
%T 7,7,4,5,6,6,7,7,8,8,8,4,5,6,6,7,7,8,8,8,7,4,5,6,6,7,7,8,8,8,8,8,4,5,
%U 7,6,7,7,8,7,8,8,8,8,4,5,8,6,7,7,8,7,8,8,8,8,8,4,5,8,6,7,7,8,8,8,8,8,8,8,8,4,5,8,6,7,7,8,8,8,8,8,8,9,9,9,4,5,7,6,7,7,8,8,8,8,8,8,9,9,9,9,4,5,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,4,5,8,7,8,8,8,8,8,8,8,8,9,9,9,10,10,9
%N Maximum number of vertices of any convex polygon formed by drawing all line segments connecting any two lattice points of an n X m convex lattice polygon in the plane written as triangle T(n,m), n >= 1, 1 <= m <= n.
%C The table is given in the section "Results" of the notes by M. E. Pfetsch and G. M. Ziegler, see link.
%C An n X m convex lattice polygon presumably means an n X m grid of square cells, formed using a grid of n+1 X m+1 points. - _N. J. A. Sloane_, Feb 07 2019
%H Huntington Tracy Hall, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.118.8423">Counterexamples in Discrete Geometry</a>. Dissertation UC, Berkeley, Fall 2004.
%H Serkan Hosten, Diane Maclagan, and Bernd Sturmfels, <a href="https://arxiv.org/abs/math/0105036">Supernormal Vector Configurations</a>, arXiv:math/0105036 [math.CO], 4 May 2001.
%H Marc E. Pfetsch and Günter M. Ziegler, <a href="http://www.mathematik.tu-darmstadt.de/~pfetsch/chambers/">Large Chambers in a Lattice Polygon</a> (Notes), March 28, 2001, December 13, 2004.
%H Marc E. Pfetsch and Günter M. Ziegler, <a href="/A288177/a288177_1.pdf">Large Chambers in a Lattice Polygon</a> (Notes), March 28, 2001, December 13, 2004. [Cached copy, with permission]
%H Hugo Pfoertner, <a href="/A288177/a288177.pdf">Illustrations of Chamber Complexes up to 5 X 5</a>.
%e Drawing the diagonals in a lattice square of size 1 X 1 produces 4 triangles, so T(1,1)=3.
%e Triangle begins:
%e 3;
%e 4, 4;
%e 4, 4, 4;
%e 4, 4, 5, 5;
%e 4, 5, 5, 6, 6;
%e 4, 5, 5, 6, 6, 6;
%e 4, 5, 6, 6, 6, 7, 7;
%e ...
%Y Cf. A288178 (diagonal of table), A288179, A288180, A288181, A288187.
%K nonn,tabl
%O 1,1
%A _Hugo Pfoertner_, Jun 06 2017
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