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A331453 Triangle read by rows: T(n,m) (n >= m >= 1) = number of vertices formed by drawing the lines connecting any two of the 2*(m+n) perimeter points of an m X n grid of squares.

%I A331453 #26 May 21 2021 07:03:58
%S A331453 5,13,37,35,99,257,75,213,421,817,159,401,881,1489,2757,275,657,1305,
%T A331453 2143,3555,4825,477,1085,2131,3431,5821,7663,12293,755,1619,2941,4817,
%U A331453 7477,9913,15037,19241,1163,2327,4369,6495,10393,13647,20425,24651,33549,1659,3257,5603,8637,13689,16953,25125,30779,39857,49577
%N A331453 Triangle read by rows: T(n,m) (n >= m >= 1) = number of vertices formed by drawing the lines connecting any two of the 2*(m+n) perimeter points of an m X n grid of squares.
%C A331453 Take a grid of m+1 X n+1 points. There are 2*(m+n) points on the perimeter. Join every pair of the perimeter points by a line (of finite length). The lines do not extend outside the grid. T(m,n) is the number of vertices in the resulting diagram, and A331452(m,n) and A331454(m,n) give the number of regions and the number of line segments respectively.
%C A331453 For illustrations see the links in A331452.
%H A331453 Lars Blomberg, <a href="/A331453/b331453.txt">Table of n, a(n) for n = 1..703</a> (the first 37 rows)
%H A331453 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 A331453 N. J. A. Sloane (in collaboration with Scott R. Shannon), <a href="/A331452/a331452.pdf">Art and Sequences</a>, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.
%H A331453 N. J. A. Sloane, <a href="https://vimeo.com/457349959">Conant's Gasket, Recamán Variations, the Enots Wolley Sequence, and Stained Glass Windows</a>, Experimental Math Seminar, Rutgers University, Sep 10 2020 (video of Zoom talk)
%e A331453 Triangle begins:
%e A331453 5,
%e A331453 13, 37,
%e A331453 35, 99, 257,
%e A331453 75, 213, 421, 817,
%e A331453 159, 401, 881, 1489, 2757,
%e A331453 275, 657, 1305, 2143, 3555, 4825,
%e A331453 477, 1085, 2131, 3431, 5821, 7663, 12293,
%e A331453 755, 1619, 2941, 4817, 7477, 9913, 15037, 19241,
%e A331453 1163, 2327, 4369, 6495, 10393, 13647, 20425, 24651, 33549,
%e A331453 ...
%Y A331453 The main diagonal is A331449.
%Y A331453 The first two columns are A331755 and A331763.
%Y A331453 Cf. A288187, A331452, A331454.
%K A331453 nonn,tabl
%O A331453 1,1
%A A331453 _Scott R. Shannon_ and _N. J. A. Sloane_, Jan 27 2020