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.

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

1,1

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. 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

Lars Blomberg, Table of n, a(n) for n = 1..703 (the first 37 rows)

Lars Blomberg, Scott R. Shannon, N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918.

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]

Scott R. Shannon, Colored illustration for T(1,1)

Scott R. Shannon, Colored illustration for T(2,1)

Scott R. Shannon, Colored illustration for T(3,1)

Scott R. Shannon, Colored illustration for T(4,1)

Scott R. Shannon, Colored illustration for T(5,1)

Scott R. Shannon, Colored illustration for T(6,1)

Scott R. Shannon, Colored illustration for T(7,1)

Scott R. Shannon, Colored illustration for T(8,1)

Scott R. Shannon, Colored illustration for T(9,1)

Scott R. Shannon, Colored illustration for T(10,1)

Scott R. Shannon, Colored illustration for T(11,1)

Scott R. Shannon, Colored illustration for T(12,1)

Scott R. Shannon, Colored illustration for T(13,1)

Scott R. Shannon, Colored illustration for T(14,1)

Scott R. Shannon, Colored illustration for T(15,1)

Scott R. Shannon, Colored illustration for T(2,2)

Scott R. Shannon, Colored illustration for T(3,2)

Scott R. Shannon, Colored illustration for T(4,2)

Scott R. Shannon, Colored illustration for T(5,2)

Scott R. Shannon, Colored illustration for T(6,2)

Scott R. Shannon, Colored illustration for T(9,2)

Scott R. Shannon, Colored illustration for T(9,2) (edge number coloring)

Scott R. Shannon, Colored illustration for T(10,2)

Scott R. Shannon, Colored illustration for T(10,2) (edge number coloring)

Scott R. Shannon, Colored illustration for T(3,3)

Scott R. Shannon, Colored illustration for T(4,3)

Scott R. Shannon, Colored illustration for T(5,3)

Scott R. Shannon, Colored illustration for T(6,3)

Scott R. Shannon, Colored illustration for T(9,3)

Scott R. Shannon, Colored illustration for T(11,3) [The top of the figure has been modified]

Scott R. Shannon, Colored illustration for T(4,4)

Scott R. Shannon, Colored illustration for T(5,4)

Scott R. Shannon, Colored illustration for T(6,4)

Scott R. Shannon, Colored illustration for T(5,5)

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(6,6) (another version)

Scott R. Shannon, Colored illustration for T(7,7)

Scott R. Shannon, Colored illustration for T(10,7)

Scott R. Shannon, Data underlying this triangle and A331453, A331454 [Includes numbers of polygonal regions with each number of edges.]

Scott R. Shannon, Data specifically for nX2 (or 2Xn) rectangles

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.

N. J. A. Sloane, Conant's Gasket, Recamán Variations, the Enots Wolley Sequence, and Stained Glass Windows, Experimental Math Seminar, Rutgers University, Sep 10 2020 (video of Zoom talk)

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

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.

Sequence in context: A223944 A127634 A331457 * A288187 A333282 A212520

Adjacent sequences: A331449 A331450 A331451 * A331453 A331454 A331455

Keyword

nonn,tabl,nice

Author

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

Status

approved