OFFSET
1,4
COMMENTS
The case m=n (the main diagonal) is dealt with in A306302, where there are illustrations for m = 1 to 15.
LINKS
M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090.
M. Griffiths, Counting the regions in a regular drawing of K_{n,n}, J. Int. Seq. 13 (2010) # 10.8.5.
S. Legendre, The Number of Crossings in a Regular Drawing of the Complete Bipartite Graph, J. Integer Seqs., Vol. 12, 2009.
Scott R. Shannon, Colored illustration for T(2,1)
Scott R. Shannon, Colored illustration for T(2,2)
Scott R. Shannon, Colored illustration for T(3,1)
Scott R. Shannon, Colored illustration for T(3,2)
Scott R. Shannon, Colored illustration for T(3,3)
Scott R. Shannon, Colored illustration for T(4,1)
Scott R. Shannon, Colored illustration for T(4,2)
Scott R. Shannon, Colored illustration for T(4,3)
Scott R. Shannon, Colored illustration for T(4,4)
Scott R. Shannon, Colored illustration for T(5,1)
Scott R. Shannon, Colored illustration for T(5,2)
Scott R. Shannon, Colored illustration for T(5,3)
Scott R. Shannon, Colored illustration for T(5,4)
Scott R. Shannon, Colored illustration for T(5,5)
Scott R. Shannon, Colored illustration for T(6,2)
Scott R. Shannon, Colored illustration for T(6,3)
Scott R. Shannon, Colored illustration for T(6,4)
Scott R. Shannon, Colored illustration for T(6,5)
Scott R. Shannon, Colored illustration for T(6,6)
Scott R. Shannon, Colored illustration for T(7,1)
Scott R. Shannon, Colored illustration for T(7,2)
Scott R. Shannon, Colored illustration for T(7,3)
Scott R. Shannon, Colored illustration for T(7,4)
Scott R. Shannon, Colored illustration for T(7,5)
Scott R. Shannon, Colored illustration for T(7,6)
Scott R. Shannon, Colored illustration for T(7,7)
Scott R. Shannon, Colored illustration for T(8,1)
Scott R. Shannon, Colored illustration for T(8,2)
Scott R. Shannon, Colored illustration for T(8,3)
Scott R. Shannon, Colored illustration for T(8,4)
Scott R. Shannon, Colored illustration for T(8,5)
Scott R. Shannon, Colored illustration for T(8,6)
Scott R. Shannon, Colored illustration for T(8,7)
Scott R. Shannon, Colored illustration for T(8,8)
Scott R. Shannon, Colored illustration for T(14,7)
FORMULA
Comment from Max Alekseyev, Jun 28 2020 (Start):
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.
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)
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)
EXAMPLE
The initial rows of the array are:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...
1, 4, 8, 13, 19, 26, 34, 43, 53, 64, 76, 89, ...
2, 8, 16, 27, 40, 56, 74, 95, 118, 144, 172, 203, ...
3, 13, 27, 46, 69, 98, 130, 168, 210, 257, 308, 365, ...
4, 19, 40, 69, 104, 149, 198, 257, 322, 395, 474, 563, ...
5, 26, 56, 98, 149, 214, 285, 371, 466, 573, 688, 818, ...
6, 34, 74, 130, 198, 285, 380, 496, 624, 768, 922, 1097, ...
7, 43, 95, 168, 257, 371, 496, 648, 816, 1005, 1207, 1437, ...
8, 53, 118, 210, 322, 466, 624, 816, 1028, 1267, 1522, 1813, ...
9, 64, 144, 257, 395, 573, 768, 1005, 1267, 1562, 1877, 2237, ...
10, 76, 172, 308, 474, 688, 922, 1207, 1522, 1877, 2256, 2690, ...
...
The initial antidiagonals are:
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
7, 34, 56, 69, 69, 56, 34, 7
8, 43, 74, 98, 104, 98, 74, 43, 8
9, 53, 95, 130, 149, 149, 130, 95, 53, 9
10, 64, 118, 168, 198, 214, 198, 168, 118, 64, 10
...
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
STATUS
approved