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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.

%I A288187 #77 Aug 06 2024 11:04:31
%S A288187 4,16,56,46,176,520,104,388,1152,2584,214,822,2502,5700,12368,380,
%T A288187 1452,4392,9944,21504,37400,648,2516,7644,17380,37572,65810,115532,
%U A288187 1028,3952,12120,27572,59784,105128,184442,294040,1562,6060,18476,42066,91654,161352,282754,450864,690816
%N 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.
%C A288187 Chambers are counted regardless of their numbers of vertices.
%C A288187 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
%H A288187 Lars Blomberg, <a href="/A288187/b288187.txt">Table of n, a(n) for n = 1..325</a> (The first 25 rows)
%H A288187 Lars Blomberg, <a href="/A288187/a288187_6.png">Colored illustration for 3 x 3</a>
%H A288187 Lars Blomberg, <a href="/A288187/a288187_7.png">Colored illustration for 4 X 4</a>
%H A288187 Lars Blomberg, <a href="/A288187/a288187_8.png">Colored illustration for 5 X 3</a>
%H A288187 Lars Blomberg, <a href="/A288187/a288187_9.png">Colored illustration for 5 X 5</a>
%H A288187 Lars Blomberg, Scott R. Shannon, N. J. A. Sloane, <a href="http://neilsloane.com/doc/rose_5.pdf">Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids</a>, (2020). Also arXiv:2009.07918.
%H A288187 Huntington Tracy Hall, <a href="https://citeseerx.ist.psu.edu/pdf/e5d8674f2073ad215df390ea390d802103ae6cea">Counterexamples in Discrete Geometry</a>. Dissertation, Department of Mathematics, University of California Berkeley, Fall 2004.
%H A288187 Serkan Hosten, Diane Maclagan, Bernd Sturmfels, <a href="https://arxiv.org/abs/math/0105036">Supernormal Vector Configurations</a>, arXiv:math/0105036 [math.CO], 4 May 2001.
%H A288187 Marc E. Pfetsch, 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 A288187 Marc E. Pfetsch, 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 A288187 Hugo Pfoertner, <a href="/A288177/a288177.pdf">Illustrations of Chamber Complexes up to 5 X 5</a>.
%e A288187 The diagonals of the 1 X 1 lattice polygon, i.e. the square, cut it into 4 triangles. Therefore T(1,1)=4.
%e A288187 Triangle begins
%e A288187 4,
%e A288187 16, 56,
%e A288187 46, 176, 520,
%e A288187 104, 388, 1152, 2584,
%e A288187 214, 822, 2502, 5700, 12368,
%e A288187 ...
%Y A288187 Cf. A288177, A288180, A288181.
%Y A288187 The first column is A306302. For column 2 see A333279, A333280, A333281.
%Y A288187 If the initial points are arranged around a circle rather than a square we get A006533 and A007678.
%K A288187 nonn,tabl
%O A288187 1,1
%A A288187 _Hugo Pfoertner_, Jun 06 2017
%E A288187 T(4,1) added from A306302. - _N. J. A. Sloane_, Feb 07 2019
%E A288187 T(3,3) corrected and rows for n=4..9 added by _Max Alekseyev_, Apr 05 2019.