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A335678 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of cells 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.

%I A335678 #39 May 29 2025 06:05:53
%S A335678 0,1,1,2,4,2,3,8,8,3,4,13,16,13,4,5,19,27,27,19,5,6,26,40,46,40,26,6,
%T A335678 7,34,56,69,69,56,34,7,8,43,74,98,104,98,74,43,8,9,53,95,130,149,149,
%U A335678 130,95,53,9,10,64,118,168,198,214,198,168,118,64,10,11,76,144,210,257,285,285,257,210,144,76,11
%N A335678 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of cells 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 A335678 The case m=n (the main diagonal) is dealt with in A306302, where there are illustrations for m = 1 to 15.
%H A335678 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 A335678 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 A335678 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 A335678 Scott R. Shannon, <a href="/A335678/a335678_1.png">Colored illustration for T(2,1)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_2.png">Colored illustration for T(2,2)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_3.png">Colored illustration for T(3,1)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_4.png">Colored illustration for T(3,2)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_5.png">Colored illustration for T(3,3)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_6.png">Colored illustration for T(4,1)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_7.png">Colored illustration for T(4,2)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_8.png">Colored illustration for T(4,3)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_9.png">Colored illustration for T(4,4)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_10.png">Colored illustration for T(5,1)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_11.png">Colored illustration for T(5,2)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_12.png">Colored illustration for T(5,3)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_13.png">Colored illustration for T(5,4)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_14.png">Colored illustration for T(5,5)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_15.png">Colored illustration for T(6,2)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_16.png">Colored illustration for T(6,3)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_17.png">Colored illustration for T(6,4)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_18.png">Colored illustration for T(6,5)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_19.png">Colored illustration for T(6,6)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_20.png">Colored illustration for T(7,1)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_21.png">Colored illustration for T(7,2)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_22.png">Colored illustration for T(7,3)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_23.png">Colored illustration for T(7,4)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_24.png">Colored illustration for T(7,5)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_25.png">Colored illustration for T(7,6)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_26.png">Colored illustration for T(7,7)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_27.png">Colored illustration for T(8,1)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_28.png">Colored illustration for T(8,2)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_29.png">Colored illustration for T(8,3)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_30.png">Colored illustration for T(8,4)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_31.png">Colored illustration for T(8,5)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_32.png">Colored illustration for T(8,6)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_33.png">Colored illustration for T(8,7)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_34.png">Colored illustration for T(8,8)</a>
%H A335678 Scott R. Shannon, <a href="/A335678/a335678_35.png">Colored illustration for T(14,7)</a>
%H A335678 <a href="/index/St#Stained">Index entries for sequences related to stained glass windows</a>
%F A335678 Euler's formula implies that A335679[m,n] = A335678[m,n] + A335680[m,n] - 1 for all m,n.
%F A335678 Comment from _Max Alekseyev_, Jun 28 2020 (Start):
%F A335678 T(m,n) = A114999(m-1,n-1) + m*n - 1 for all m, n >= 1. This follows from the Alekseyev-Basova-Zolotykh (2015) article.
%F A335678 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)
%F A335678 _Max Alekseyev_'s formula is an analog of Theorem 3 of Griffiths (2010), and gives an explicit formula for this array. - _N. J. A. Sloane_, Jun 30 2020
%e A335678 The initial rows of the array are:
%e A335678   0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...
%e A335678   1, 4, 8, 13, 19, 26, 34, 43, 53, 64, 76, 89, ...
%e A335678   2, 8, 16, 27, 40, 56, 74, 95, 118, 144, 172, 203, ...
%e A335678   3, 13, 27, 46, 69, 98, 130, 168, 210, 257, 308, 365, ...
%e A335678   4, 19, 40, 69, 104, 149, 198, 257, 322, 395, 474, 563, ...
%e A335678   5, 26, 56, 98, 149, 214, 285, 371, 466, 573, 688, 818, ...
%e A335678   6, 34, 74, 130, 198, 285, 380, 496, 624, 768, 922, 1097, ...
%e A335678   7, 43, 95, 168, 257, 371, 496, 648, 816, 1005, 1207, 1437, ...
%e A335678   8, 53, 118, 210, 322, 466, 624, 816, 1028, 1267, 1522, 1813, ...
%e A335678   9, 64, 144, 257, 395, 573, 768, 1005, 1267, 1562, 1877, 2237, ...
%e A335678   10, 76, 172, 308, 474, 688, 922, 1207, 1522, 1877, 2256, 2690, ...
%e A335678   ...
%e A335678 The initial antidiagonals are:
%e A335678   0
%e A335678   1, 1
%e A335678   2, 4, 2
%e A335678   3, 8, 8, 3
%e A335678   4, 13, 16, 13, 4
%e A335678   5, 19, 27, 27, 19, 5
%e A335678   6, 26, 40, 46, 40, 26, 6
%e A335678   7, 34, 56, 69, 69, 56, 34, 7
%e A335678   8, 43, 74, 98, 104, 98, 74, 43, 8
%e A335678   9, 53, 95, 130, 149, 149, 130, 95, 53, 9
%e A335678   10, 64, 118, 168, 198, 214, 198, 168, 118, 64, 10
%e A335678   ...
%Y A335678 This is one of a set of five arrays: A335678, A335679, A335680, A335681, A335682.
%Y A335678 For the diagonal see A306302.
%Y A335678 See also A114999.
%K A335678 nonn,tabl
%O A335678 1,4
%A A335678 _Lars Blomberg_, _Scott R. Shannon_, and _N. J. A. Sloane_, Jun 28 2020