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.
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, 48, 13, 15, 63, 97, 119, 119, 97, 63, 15, 17, 80, 128, 170, 178, 170, 128, 80, 17, 19, 99, 165, 225, 257, 257, 225, 165, 99, 19, 21, 120, 205, 292, 340, 372, 340, 292, 205, 120, 21
Offset: 1
Examples
The initial rows of the array are: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, ... 3, 8, 15, 24, 35, 48, 63, 80, 99, 120, 143, 168, ... 5, 15, 28, 47, 69, 97, 128, 165, 205, 251, 300, 355, ... 7, 24, 47, 80, 119, 170, 225, 292, 365, 448, 537, 638, ... 9, 35, 69, 119, 178, 257, 340, 443, 555, 683, 819, 975, ... 11, 48, 97, 170, 257, 372, 493, 644, 809, 998, 1197, 1426, ... 13, 63, 128, 225, 340, 493, 654, 857, 1078, 1331, 1595, 1901, ... 15, 80, 165, 292, 443, 644, 857, 1124, 1415, 1748, 2095, 2498, ... 17, 99, 205, 365, 555, 809, 1078, 1415, 1782, 2203, 2640, 3149, ... 19, 120, 251, 448, 683, 998, 1331, 1748, 2203, 2724, 3265, 3896, ... 21, 143, 300, 537, 819, 1197, 1595, 2095, 2640, 3265, 3914, 4673, ... ... The initial antidiagonals are: 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, 48, 13 15, 63, 97, 119, 119, 97, 63, 15 17, 80, 128, 170, 178, 170, 128, 80, 17 19, 99, 165, 225, 257, 257, 225, 165, 99, 19 21, 120, 205, 292, 340, 372, 340, 292, 205, 120, 21 23, 143, 251, 365, 443, 493, 493, 443, 365, 251, 143, 23 ...
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.
- Index entries for sequences related to stained glass windows
Crossrefs
Formula
Comment from Max Alekseyev, Jun 28 2020 (Start):
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.
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)
Comments