

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.


99



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
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OFFSET

1,1


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.


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. 6597.
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. 6597.


LINKS

Johnny Fonseca, Intersections and Segments, Illustrations for T(n,m) with 2 <= n <= m <= 10, with intersection points shown on the left, and the full structures on the right. Solution to homework problem, Math 640, Rutgers Univ., Feb 11 2020.
Johnny Fonseca, Intersections and Segments, Illustrations for T(n,m) with 2 <= n <= m <= 10, with intersection points shown on the left, and the full structures on the right. Solution to homework problem, Math 640, Rutgers Univ., Feb 11 2020. [Local copy]
N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.


EXAMPLE

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;
...


CROSSREFS

See A333274 for the classification of vertices by valency.


KEYWORD



AUTHOR



STATUS

approved



