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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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.

Original entry on oeis.org

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
Offset: 1

Views

Author

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

Keywords

Comments

The case m=n (the main diagonal) is dealt with in A306302, where there are illustrations for m = 1 to 15.

Examples

			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
  ...

Links

Crossrefs

This is one of a set of five arrays: A335678, A335679, A335680, A335681, A335682.
For the diagonal see A306302.
See also A114999.

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

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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