A335680 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of vertices 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.

2, 3, 3, 4, 5, 4, 5, 8, 8, 5, 6, 12, 13, 12, 6, 7, 17, 21, 21, 17, 7, 8, 23, 30, 35, 30, 23, 8, 9, 30, 42, 51, 51, 42, 30, 9, 10, 38, 55, 73, 75, 73, 55, 38, 10, 11, 47, 71, 96, 109, 109, 96, 71, 47, 11, 12, 57, 88, 125, 143, 159, 143, 125, 88, 57, 12, 13, 68, 108, 156, 187, 209, 209, 187, 156, 108, 68, 13

Offset

1,1

Comments

The case m=n (the main diagonal) is dealt with in A331755. A306302 has illustrations for the diagonal case for m = 1 to 15.

Also A335678 has colored illustrations for many values of m and n.

Links

Table of n, a(n) for n=1..78.

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

Formula

Comment from Max Alekseyev, Jun 28 2020 (Start):

T(m,n) = A114999(m-1,n-1) - A331762(m-1,n-1) + m + n 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 Proposition 9 of Legendre (2009), and gives an explicit formula for this array. - N. J. A. Sloane, Jun 30 2020)

Example

The initial rows of the array are:
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, ...
3, 5, 8, 12, 17, 23, 30, 38, 47, 57, 68, 80, ...
4, 8, 13, 21, 30, 42, 55, 71, 88, 108, 129, 153, ...
5, 12, 21, 35, 51, 73, 96, 125, 156, 192, 230, 274, ...
6, 17, 30, 51, 75, 109, 143, 187, 234, 289, 346, 413, ...
7, 23, 42, 73, 109, 159, 209, 274, 344, 426, 510, 609, ...
8, 30, 55, 96, 143, 209, 275, 362, 455, 564, 674, 805, ...
9, 38, 71, 125, 187, 274, 362, 477, 600, 744, 889, 1062, ...
10, 47, 88, 156, 234, 344, 455, 600, 755, 937, 1119, 1337, ...
11, 57, 108, 192, 289, 426, 564, 744, 937, 1163, 1389, 1660, ...
12, 68, 129, 230, 346, 510, 674, 889, 1119, 1389, 1659, 1984, ...
...
The initial antidiagonals are:
2
3, 3
4, 5, 4
5, 8, 8, 5
6, 12, 13, 12, 6
7, 17, 21, 21, 17, 7
8, 23, 30, 35, 30, 23, 8
9, 30, 42, 51, 51, 42, 30, 9
10, 38, 55, 73, 75, 73, 55, 38, 10
11, 47, 71, 96, 109, 109, 96, 71, 47, 11
12, 57, 88, 125, 143, 159, 143, 125, 88, 57, 12
...

Crossrefs

This is one of a set of five arrays: A335678, A335679, A335680, A335681, A335682.

For the diagonal case see A306302 and A331755.

Cf. A114999, A331762.

Sequence in context: A115729 A115728 A188553 * A026354 A179840 A357431

Adjacent sequences: A335677 A335678 A335679 * A335681 A335682 A335683

Keyword

nonn,tabl

Author

Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Jun 28 2020

Status

approved