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A331454 Triangle read by rows: T(n,m) (n >= m >= 1) = number of line segments formed by drawing the lines connecting any two of the 2*(m+n) perimeter points of an m X n grid of squares.

%I A331454 #21 May 23 2020 09:04:11
%S A331454 8,28,92,80,240,596,178,508,1028,1936,372,944,2004,3404,6020,654,1548,
%T A331454 3018,4962,8064,11088,1124,2520,4808,7734,12708,17022,26260,1782,3754,
%U A331454 6704,10840,16608,22220,32794,42144,2724,5392,9780,14620,22788,30238,44028,54024,72296,3914,7528,12720,19428,29914,37848,54612,67590,86906,107832
%N A331454 Triangle read by rows: T(n,m) (n >= m >= 1) = number of line segments formed by drawing the lines connecting any two of the 2*(m+n) perimeter points of an m X n grid of squares.
%C A331454 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 line segments formed when these lines intersect each other, and A331452(m,n) and A331453(m,n) give the number of regions and the number of vertices respectively.
%C A331454 For illustrations see the links in A331452.
%H A331454 Lars Blomberg, <a href="/A331454/b331454.txt">Table of n, a(n) for n = 1..703</a> (the first 37 rows)
%H A331454 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.
%e A331454 Triangle begins:
%e A331454 8,
%e A331454 28, 92,
%e A331454 80, 240, 596,
%e A331454 178, 508, 1028, 1936,
%e A331454 372, 944, 2004, 3404, 6020,
%e A331454 654, 1548, 3018, 4962, 8064, 11088,
%e A331454 1124, 2520, 4808, 7734, 12708, 17022, 26260,
%e A331454 1782, 3754, 6704, 10840, 16608, 22220, 32794, 42144,
%e A331454 2724, 5392, 9780, 14620, 22788, 30238, 44028, 54024, 72296,
%e A331454 ...
%Y A331454 The main diagonal is A331448.
%Y A331454 The first two columns are A331757 and A331765.
%Y A331454 Cf. A288187, A331452, A331453.
%K A331454 nonn,tabl
%O A331454 1,1
%A A331454 _Scott R. Shannon_ and _N. J. A. Sloane_, Jan 27 2020