This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A335679 #22 Jun 30 2020 20:52:09
%S A335679 1,3,3,5,8,5,7,15,15,7,9,24,28,24,9,11,35,47,47,35,11,13,48,69,80,69,
%T A335679 48,13,15,63,97,119,119,97,63,15,17,80,128,170,178,170,128,80,17,19,
%U A335679 99,165,225,257,257,225,165,99,19,21,120,205,292,340,372,340,292,205,120,21
%N A335679 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of edges in figure formed by taking m equally spaced points on a line and n equally spaced points on a parallel line, and joining each of the m points to each of the n points by a line segment.
%C A335679 The case m=n (the main diagonal) is dealt with in A331757. A306302 has illustrations for the diagonal case for m = 1 to 15.
%C A335679 Also A335678 has colored illustrations for many values of m and n.
%H A335679 M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. <a href="https://doi.org/10.1137/140978090">On the minimal teaching sets of two-dimensional threshold functions</a>. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090.
%H A335679 M. Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Griffiths2/griffiths.html">Counting the regions in a regular drawing of K_{n,n}</a>, J. Int. Seq. 13 (2010) # 10.8.5.
%H A335679 S. Legendre, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL12/Legendre/legendre2.html">The Number of Crossings in a Regular Drawing of the Complete Bipartite Graph</a>, J. Integer Seqs., Vol. 12, 2009.
%H A335679 <a href="/index/St#Stained">Index entries for sequences related to stained glass windows</a>
%F A335679 Comment from _Max Alekseyev_, Jun 28 2020 (Start):
%F A335679 T(m,n) = 2*A114999(m-1,n-1) - A331762(m-1,n-1) + m*n + m + n - 2 for all m, n >= 1. This follows from the Alekseyev-Basova-Zolotykh (2015) article.
%F A335679 Proof: Here is the appropriate modification of the corresponding comment in A306302: Assuming that K(m,n) has vertices at (i,0) and (j,1), for i=0..m-1 and j=0..n-1, the projective map (x,y) -> ((1-y)/(x+1), y/(x+1)) maps K(m,n) to the partition of the right triangle described by Alekseyev et al. (2015), for which Theorem 13 gives the number of regions, line segments, and intersection points. (End)
%e A335679 The initial rows of the array are:
%e A335679 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, ...
%e A335679 3, 8, 15, 24, 35, 48, 63, 80, 99, 120, 143, 168, ...
%e A335679 5, 15, 28, 47, 69, 97, 128, 165, 205, 251, 300, 355, ...
%e A335679 7, 24, 47, 80, 119, 170, 225, 292, 365, 448, 537, 638, ...
%e A335679 9, 35, 69, 119, 178, 257, 340, 443, 555, 683, 819, 975, ...
%e A335679 11, 48, 97, 170, 257, 372, 493, 644, 809, 998, 1197, 1426, ...
%e A335679 13, 63, 128, 225, 340, 493, 654, 857, 1078, 1331, 1595, 1901, ...
%e A335679 15, 80, 165, 292, 443, 644, 857, 1124, 1415, 1748, 2095, 2498, ...
%e A335679 17, 99, 205, 365, 555, 809, 1078, 1415, 1782, 2203, 2640, 3149, ...
%e A335679 19, 120, 251, 448, 683, 998, 1331, 1748, 2203, 2724, 3265, 3896, ...
%e A335679 21, 143, 300, 537, 819, 1197, 1595, 2095, 2640, 3265, 3914, 4673, ...
%e A335679 ...
%e A335679 The initial antidiagonals are:
%e A335679 1
%e A335679 3, 3
%e A335679 5, 8, 5
%e A335679 7, 15, 15, 7
%e A335679 9, 24, 28, 24, 9
%e A335679 11, 35, 47, 47, 35, 11
%e A335679 13, 48, 69, 80, 69, 48, 13
%e A335679 15, 63, 97, 119, 119, 97, 63, 15
%e A335679 17, 80, 128, 170, 178, 170, 128, 80, 17
%e A335679 19, 99, 165, 225, 257, 257, 225, 165, 99, 19
%e A335679 21, 120, 205, 292, 340, 372, 340, 292, 205, 120, 21
%e A335679 23, 143, 251, 365, 443, 493, 493, 443, 365, 251, 143, 23
%e A335679 ...
%Y A335679 This is one of a set of five arrays: A335678, A335679, A335680, A335681, A335682.
%Y A335679 For the diagonal case see A306302 and A331757.
%Y A335679 Cf. A114999, A331762.
%K A335679 nonn,tabl
%O A335679 1,2
%A A335679 _Lars Blomberg_, _Scott R. Shannon_, and _N. J. A. Sloane_, Jun 28 2020