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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.

A331452 Triangle read by rows: T(n,m) (n >= m >= 1) = number of regions (or cells) formed by drawing the line segments connecting any two of the 2*(m+n) perimeter points of an m X n grid of squares.

Original entry on oeis.org

4, 16, 56, 46, 142, 340, 104, 296, 608, 1120, 214, 544, 1124, 1916, 3264, 380, 892, 1714, 2820, 4510, 6264, 648, 1436, 2678, 4304, 6888, 9360, 13968, 1028, 2136, 3764, 6024, 9132, 12308, 17758, 22904, 1562, 3066, 5412, 8126, 12396, 16592, 23604, 29374, 38748, 2256, 4272, 7118, 10792, 16226, 20896, 29488, 36812, 47050, 58256
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Jan 27 2020

Keywords

Comments

Take a grid of m+1 X n+1 points. There are 2*(m+n) points on the perimeter. Join every pair of the perimeter points by a line segment. The lines do not extend outside the grid. T(m,n) is the number of regions formed by these lines, and A331453(m,n) and A331454(m,n) give the number of vertices and the number of line segments respectively.
A288187 is a similar sequence, except there every pair of the (m+1)*(n+1) points of the grid (including the interior points) are joined by line segments. The (m,1) (m>=1) and (2,2) entries here and in A288187 are the same, while all other entries are different.

Examples

			Triangle begins:
     4;
    16,   56;
    46,  142,  340;
   104,  296,  608,  1120;
   214,  544, 1124,  1916,  3264;
   380,  892, 1714,  2820,  4510,  6264;
   648, 1436, 2678,  4304,  6888,  9360, 13968;
  1028, 2136, 3764,  6024,  9132, 12308, 17758, 22904;
  1562, 3066, 5412,  8126, 12396, 16592, 23604, 29374, 38748;
  2256, 4272, 7118, 10792, 16226, 20896, 29488, 36812, 47050, 58256;
  ...

References

  • Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, Integers, Ron Graham Memorial Volume 21A (2021), #A5. Also in book, "Number Theory and Combinatorics: A Collection in Honor of the Mathematics of Ronald Graham", ed. B. M. Landman et al., De Gruyter, 2022, pp. 65-97.
  • Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, Integers, Ron Graham Memorial Volume 21A (2021), #A5. Also in book, "Number Theory and Combinatorics: A Collection in Honor of the Mathematics of Ronald Graham", ed. B. M. Landman et al., De Gruyter, 2022, pp. 65-97.

Links

Crossrefs

The first column is A306302, the main diagonal is A255011.
The second column is A331766.
See A333274 for the classification of vertices by valency.
Cf. A288187, A331453, A331454, A333286, A333287, A333288.

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

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