A288181 -id:A288181 - cp's OEIS frontend

A288187 Triangle read by rows: T(n,m) (n >= m >= 1) = number of chambers (or regions) formed by drawing the line segments connecting any two of the (n+1) X (m+1) lattice points in an n X m lattice polygon.

4, 16, 56, 46, 176, 520, 104, 388, 1152, 2584, 214, 822, 2502, 5700, 12368, 380, 1452, 4392, 9944, 21504, 37400, 648, 2516, 7644, 17380, 37572, 65810, 115532, 1028, 3952, 12120, 27572, 59784, 105128, 184442, 294040, 1562, 6060, 18476, 42066, 91654, 161352, 282754, 450864, 690816

Offset: 1

Hugo Pfoertner, Jun 06 2017

Chambers are counted regardless of their numbers of vertices.

The n X m lattice polygon mentioned in the definition is 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

			The diagonals of the 1 X 1 lattice polygon, i.e. the square, cut it into 4 triangles. Therefore T(1,1)=4.
Triangle begins
4,
16, 56,
46, 176, 520,
104, 388, 1152, 2584,
214, 822, 2502, 5700, 12368,
...

Lars Blomberg, Table of n, a(n) for n = 1..325 (The first 25 rows)
Lars Blomberg, Colored illustration for 3 x 3
Lars Blomberg, Colored illustration for 4 X 4
Lars Blomberg, Colored illustration for 5 X 3
Lars Blomberg, Colored illustration for 5 X 5
Lars Blomberg, Scott R. Shannon, N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918.
Huntington Tracy Hall, Counterexamples in Discrete Geometry. Dissertation, Department of Mathematics, University of California 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.

Cf. A288177, A288180, A288181.
The first column is A306302. For column 2 see A333279, A333280, A333281.
If the initial points are arranged around a circle rather than a square we get A006533 and A007678.

T(4,1) added from A306302. - N. J. A. Sloane, Feb 07 2019

T(3,3) corrected and rows for n=4..9 added by Max Alekseyev, Apr 05 2019.

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.

Original entry on oeis.org

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

Offset: 1

Hugo Pfoertner, Jun 06 2017

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

			Drawing the diagonals in a lattice square of size 1 X 1 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;
  ...

Huntington Tracy Hall, Counterexamples in Discrete Geometry. Dissertation UC, Berkeley, Fall 2004.
Serkan Hosten, Diane Maclagan, and Bernd Sturmfels, Supernormal Vector Configurations, arXiv:math/0105036 [math.CO], 4 May 2001.
Marc E. Pfetsch and Günter M. Ziegler, Large Chambers in a Lattice Polygon (Notes), March 28, 2001, December 13, 2004.
Marc E. Pfetsch and 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.

Cf. A288178 (diagonal of table), A288179, A288180, A288181, A288187.

cp's OEIS Frontend

A288187 Triangle read by rows: T(n,m) (n >= m >= 1) = number of chambers (or regions) formed by drawing the line segments connecting any two of the (n+1) X (m+1) lattice points in an n X m lattice polygon.

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