A335680 - OEIS

%I #22 Jun 30 2020 20:54:07

%S 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,

%T 9,30,42,51,51,42,30,9,10,38,55,73,75,73,55,38,10,11,47,71,96,109,109,

%U 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

%N 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.

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

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

%H 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 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 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 <a href="/index/St#Stained">Index entries for sequences related to stained glass windows</a>

%F Comment from _Max Alekseyev_, Jun 28 2020 (Start):

%F 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.

%F 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 _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)

%e The initial rows of the array are:

%e 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, ...

%e 3, 5, 8, 12, 17, 23, 30, 38, 47, 57, 68, 80, ...

%e 4, 8, 13, 21, 30, 42, 55, 71, 88, 108, 129, 153, ...

%e 5, 12, 21, 35, 51, 73, 96, 125, 156, 192, 230, 274, ...

%e 6, 17, 30, 51, 75, 109, 143, 187, 234, 289, 346, 413, ...

%e 7, 23, 42, 73, 109, 159, 209, 274, 344, 426, 510, 609, ...

%e 8, 30, 55, 96, 143, 209, 275, 362, 455, 564, 674, 805, ...

%e 9, 38, 71, 125, 187, 274, 362, 477, 600, 744, 889, 1062, ...

%e 10, 47, 88, 156, 234, 344, 455, 600, 755, 937, 1119, 1337, ...

%e 11, 57, 108, 192, 289, 426, 564, 744, 937, 1163, 1389, 1660, ...

%e 12, 68, 129, 230, 346, 510, 674, 889, 1119, 1389, 1659, 1984, ...

%e ...

%e The initial antidiagonals are:

%e 2

%e 3, 3

%e 4, 5, 4

%e 5, 8, 8, 5

%e 6, 12, 13, 12, 6

%e 7, 17, 21, 21, 17, 7

%e 8, 23, 30, 35, 30, 23, 8

%e 9, 30, 42, 51, 51, 42, 30, 9

%e 10, 38, 55, 73, 75, 73, 55, 38, 10

%e 11, 47, 71, 96, 109, 109, 96, 71, 47, 11

%e 12, 57, 88, 125, 143, 159, 143, 125, 88, 57, 12

%e ...

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

%Y For the diagonal case see A306302 and A331755.

%Y Cf. A114999, A331762.

%K nonn,tabl

%O 1,1

%A _Lars Blomberg_, _Scott R. Shannon_, and _N. J. A. Sloane_, Jun 28 2020