

A288177


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.


10



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, 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, 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
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OFFSET

1,1


COMMENTS

The table is given in the section "Results" of the notes by M. E. Pfetsch and G. M. Ziegler, see link.
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


LINKS

Table of n, a(n) for n=1..171.
Huntington Tracy Hall, Counterexamples in Discrete Geometry. Dissertation UC, Berkeley, Fall 2004.
Serkan Hosten, Diane Maclagan, Bernd Sturmfels, Supernormal Vector Configurations, arXiv:math/0105036 [math.CO], 4 May 2001.
Marc E. Pfetsch, Günter M. Ziegler, Large Chambers in a Lattice Polygon (Notes), March 28, 2001, December 13, 2004.
Marc E. Pfetsch, Günter M. Ziegler, Large Chambers in a Lattice Polygon (Notes), March 28, 2001, December 13, 2004. [Cached copy, with permission]
Hugo Pfoertner, Illustrations of Chamber Complexes up to 5 X 5.


EXAMPLE

Drawing the diagonals in a lattice square of size 1X1 produces 4 triangles, so T(1,1)=3.
Triangle begins:
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,
...


CROSSREFS

Cf. A288178 (diagonal of table), A288179, A288180, A288181, A288187.
Sequence in context: A018244 A113311 A255176 * A064042 A332325 A194882
Adjacent sequences: A288174 A288175 A288176 * A288178 A288179 A288180


KEYWORD

nonn,tabl


AUTHOR

Hugo Pfoertner, Jun 06 2017


STATUS

approved



