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A333282 Triangle read by rows: T(m,n) (m >= n >= 1) = number of regions formed by drawing the line segments connecting any two of the (m+1) X (n+1) lattice points in an m X n grid of squares and extending them to the boundary of the grid.

%I A333282 #102 May 27 2021 06:25:54
%S A333282 4,16,56,46,192,624,104,428,1416,3288,214,942,3178,7520,16912,380,
%T A333282 1672,5612,13188,29588,51864,648,2940,9926,23368,52368,92518,164692,
%U A333282 1028,4624,15732,37184,83628,148292,263910,422792,1562,7160,24310,57590,130034,230856,410402,658080,1023416
%N A333282 Triangle read by rows: T(m,n) (m >= n >= 1) = number of regions formed by drawing the line segments connecting any two of the (m+1) X (n+1) lattice points in an m X n grid of squares and extending them to the boundary of the grid.
%C A333282 Triangle gives number of nodes in graph LC(m,n) in the notation of Blomberg-Shannon-Sloane (2020).
%C A333282 If we only joined pairs of the 2(m+n) boundary points, we would get A331452. If we did not extend the lines to the boundary of the grid, we would get A288187. (One of the links below shows the difference between the three definitions in the case m=3, n=2.)
%H A333282 Lars Blomberg, Scott R. Shannon, and 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 A333282 Seppo Mustonen, <a href="http://www.survo.fi/papers/GeomAccuracy.pdf">Statistical accuracy of geometric constructions</a>, 2008.
%H A333282 Seppo Mustonen, <a href="/A333282/a333282_1.pdf">Statistical accuracy of geometric constructions</a>, 2008 [Local copy]
%H A333282 Seppo Mustonen, <a href="http://www.survo.fi/papers/PointsInGrid.pdf">On lines and their intersection points in a rectangular grid of points</a>, 2009
%H A333282 Seppo Mustonen, <a href="/A018808/a018808.pdf">On lines and their intersection points in a rectangular grid of points</a>, 2009 [Local copy]
%H A333282 Seppo Mustonen, <a href="http://www.survo.fi/papers/LinesInGrid2.pdf">On lines going through a given number of points in a rectangular grid of points</a>, 2010
%H A333282 Seppo Mustonen, <a href="/A141255/a141255.pdf">On lines going through a given number of points in a rectangular grid of points</a>, 2010 [Local copy]
%H A333282 Scott R. Shannon, <a href="/A333282/a333282.png">Colored illustration for T(3,2)</a>
%H A333282 Scott R. Shannon, <a href="/A333282/a333282_1.png">Colored illustration for T(3,2)</a> (edge number coloring)
%H A333282 Scott R. Shannon, <a href="/A333282/a333282_2.png">Colored illustration for T(3,3)</a>
%H A333282 Scott R. Shannon, <a href="/A333282/a333282_3.png">Colored illustration for T(3,3)</a> (edge number coloring)
%H A333282 Scott R. Shannon, <a href="/A333282/a333282_4.png">Colored illustration for T(4,4)</a>
%H A333282 Scott R. Shannon, <a href="/A333282/a333282_5.png">Colored illustration for T(4,4)</a> (edge number coloring)
%H A333282 Scott R. Shannon, <a href="/A333282/a333282_6.png">Colored illustration for T(5,5)</a>
%H A333282 Scott R. Shannon, <a href="/A333282/a333282_7.png">Colored illustration for T(5,5)</a> (edge number coloring)
%H A333282 Scott R. Shannon, <a href="/A333282/a333282_8.png">Colored illustration for T(6,3)</a>
%H A333282 Scott R. Shannon, <a href="/A333282/a333282_9.png">Colored illustration for T(6,3)</a> (edge number coloring)
%H A333282 Scott R. Shannon, <a href="/A333282/a333282_10.png">Colored illustration for T(7,4)</a>
%H A333282 Scott R. Shannon, <a href="/A333282/a333282_11.png">Colored illustration for T(7,4)</a> (edge number coloring)
%H A333282 Scott R. Shannon, <a href="/A333282/a333282_12.png">Colored illustration for T(8,2)</a>
%H A333282 Scott R. Shannon, <a href="/A333282/a333282_13.png">Colored illustration for T(8,2)</a> (edge number coloring)
%H A333282 Scott R. Shannon, <a href="/A333282/a333282.dat.txt">Numerical properties of these structures</a>. Corrected version Mar 24 2020.
%H A333282 N. J. A. Sloane, <a href="/A333282/a333282.pdf">Illustration of T(3,2) = 192.</a> [Black lines correspond to A331452(3,2), black + red lines correspond to A288187(3,2), and black + red + blue lines to T(3,2)]
%H A333282 N. J. A. Sloane, <a href="/A333282/a333282_2.pdf">Illustration of T(3,3) = 624</a> [Black lines correspond to A288187(3,3), and black + red lines to T(3,3)]
%e A333282 Triangle begins:
%e A333282 4,
%e A333282 16, 56,
%e A333282 46, 192, 624,
%e A333282 104, 428, 1416, 3288,
%e A333282 214, 942, 3178, 7520, 16912,
%e A333282 380, 1672, 5612, 13188, 29588, 51864,
%e A333282 648, 2940, 9926, 23368, 52368, 92518, 164692,
%e A333282 1028, 4624, 15732, 37184, 83628, 148292, 263910, 422792
%e A333282 1562, 7160, 24310, 57590, 130034, 230856, 410402, 658080, 1023416
%e A333282 2256, 10336, 35132, 83116, 187376, 331484, 588618, 942808, 1466056, 2101272
%Y A333282 Cf. A288187, A331452, A333283 (edges), A333284 (vertices). Column 1 is A306302.  Main diagonal is A333294.
%K A333282 nonn,tabl
%O A333282 1,1
%A A333282 _Scott R. Shannon_ and _N. J. A. Sloane_, Mar 16 2020
%E A333282 More terms and corrections from _Scott R. Shannon_, Mar 21 2020
%E A333282 More terms from _Scott R. Shannon_, May 27 2021