A335678 - OEIS

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.

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, 11, 76, 144, 210, 257, 285, 285, 257, 210, 144, 76, 11 (list; table; graph; refs; listen; history; text; internal format)

	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	`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.` `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)` `Index entries for sequences related to stained glass windows`
	FORMULA	`Euler's formula implies that A335679[m,n] = A335678[m,n] + A335680[m,n] - 1 for all m,n.` `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	`This is one of a set of five arrays: A335678, A335679, A335680, A335681, A335682.` `For the diagonal see A306302.` `See also A114999.` `Sequence in context: A360373 A141387 A349400 * A368434 A134400 A016095` `Adjacent sequences: A335675 A335676 A335677 * A335679 A335680 A335681`
	KEYWORD	`nonn,tabl`
	AUTHOR	`Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Jun 28 2020`
	STATUS	`approved`