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A333284 Triangle read by rows: T(m,n) (m >= n >= 1) = number of vertices 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 A333284 #29 May 21 2021 07:15:40
%S A333284 5,13,37,35,129,405,75,289,933,2225,159,663,2155,5157,11641,275,1163,
%T A333284 3793,9051,20341,35677,477,2069,6771,16129,36173,63987,114409,755,
%U A333284 3251,10727,25635,57759,102845,183961
%N A333284 Triangle read by rows: T(m,n) (m >= n >= 1) = number of vertices 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 A333284 If we only joined pairs of the 2(m+n) boundary points, we would get A331453. If we did not extend the lines to the boundary of the grid, we would get A288180. (One of the links below shows the difference between the three definitions in the case m=3, n=2.)
%C A333284 See A333282 for a large number of colored illustrations.
%H A333284 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>, (2021). Also arXiv:2009.07918.
%H A333284 Seppo Mustonen, <a href="http://www.survo.fi/papers/GeomAccuracy.pdf">Statistical accuracy of geometric constructions</a>, 2008.
%H A333284 Seppo Mustonen, <a href="/A333282/a333282_1.pdf">Statistical accuracy of geometric constructions</a>, 2008 [Local copy]
%H A333284 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 A333284 Seppo Mustonen, <a href="/A018808/a018808.pdf">On lines and their intersection points in a rectangular grid of points</a>, 2009 [Local copy]
%H A333284 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 A333284 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 A333284 N. J. A. Sloane, <a href="/A333282/a333282.pdf">Illustration of T(3,2) = 129.</a> [Black lines correspond to A331453(3,2), black + red lines correspond to A288180(3,2), and black + red + blue lines to T(3,2)]
%H A333284 N. J. A. Sloane, <a href="/A333282/a333282_2.pdf">Illustration of T(3,3) = 405</a> [Black lines correspond to A288180(3,3), and black + red lines to T(3,3)]
%e A333284 Triangle begins:
%e A333284 5,
%e A333284 13, 37,
%e A333284 35, 129, 405,
%e A333284 75, 289, 933, 2225,
%e A333284 159, 663, 2155, 5157, 11641,
%e A333284 275, 1163, 3793, 9051, 20341, 35677,
%e A333284 477, 2069, 6771, 16129, 36173, 63987, 114409,
%e A333284 755, 3251, 10727, 25635, 57759, 102845, 183961, ...
%e A333284 ...
%e A333284 T(7,7) corrected Mar 19 2020
%Y A333284 Cf. A288187, A331452, A288180, A331453, A333282 (regions), A333283 (edges). Column 1 is A331755. The main diagonal is A333285.
%K A333284 nonn,tabl,more
%O A333284 1,1
%A A333284 _Scott R. Shannon_ and _N. J. A. Sloane_, Mar 16 2020
%E A333284 More terms and corrections from _Scott R. Shannon_, Mar 21 2020