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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.

%I A331452 #140 Jan 08 2023 01:07:48
%S A331452 4,16,56,46,142,340,104,296,608,1120,214,544,1124,1916,3264,380,892,
%T A331452 1714,2820,4510,6264,648,1436,2678,4304,6888,9360,13968,1028,2136,
%U A331452 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
%N 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.
%C A331452 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.
%C A331452 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.
%D A331452 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.
%D A331452 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.
%H A331452 Lars Blomberg, <a href="/A331452/b331452.txt">Table of n, a(n) for n = 1..703</a> (the first 37 rows)
%H A331452 Lars Blomberg, Scott R. Shannon, N. J. A. Sloane, <a href="http://neilsloane.com/doc/rose_5.pdf">Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids</a>, (2020). Also arXiv:2009.07918.
%H A331452 Johnny Fonseca, <a href="https://sites.math.rutgers.edu/~zeilberg/EM20/hw5/Intersections%20and%20Segments.pdf">Intersections and Segments</a>, 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.
%H A331452 Johnny Fonseca, <a href="/A331452/a331452_1.pdf">Intersections and Segments</a>, 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]
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_6.png">Colored illustration for T(1,1)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_7.png">Colored illustration for T(2,1)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_8.png">Colored illustration for T(3,1)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_9.png">Colored illustration for T(4,1)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_10.png">Colored illustration for T(5,1)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_11.png">Colored illustration for T(6,1)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_32.png">Colored illustration for T(7,1)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_33.png">Colored illustration for T(8,1)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_34.png">Colored illustration for T(9,1)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_35.png">Colored illustration for T(10,1)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_36.png">Colored illustration for T(11,1)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_37.png">Colored illustration for T(12,1)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_38.png">Colored illustration for T(13,1)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_39.png">Colored illustration for T(14,1)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_40.png">Colored illustration for T(15,1)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_12.png">Colored illustration for T(2,2)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_13.png">Colored illustration for T(3,2)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_14.png">Colored illustration for T(4,2)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_15.png">Colored illustration for T(5,2)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_16.png">Colored illustration for T(6,2)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_28.png">Colored illustration for T(9,2)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_29.png">Colored illustration for T(9,2)</a> (edge number coloring)
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_30.png">Colored illustration for T(10,2)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_31.png">Colored illustration for T(10,2)</a> (edge number coloring)
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_1.png">Colored illustration for T(3,3)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_18.png">Colored illustration for T(4,3)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_19.png">Colored illustration for T(5,3)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_20.png">Colored illustration for T(6,3)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452.png">Colored illustration for T(9,3)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_5.png">Colored illustration for T(11,3)</a> [The top of the figure has been modified]
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_21.png">Colored illustration for T(4,4)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_22.png">Colored illustration for T(5,4)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_23.png">Colored illustration for T(6,4)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_24.png">Colored illustration for T(5,5)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_25.png">Colored illustration for T(6,5)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_2.png">Colored illustration for T(6,6)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_3.png">Colored illustration for T(6,6)</a> (another version)
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_4.png">Colored illustration for T(7,7)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_27.png">Colored illustration for T(10,7)</a>
%H A331452 Scott R. Shannon, <a href="/A331452/a331452.txt">Data underlying this triangle and A331453, A331454</a> [Includes numbers of polygonal regions with each number of edges.]
%H A331452 Scott R. Shannon, <a href="/A331452/a331452_1.txt">Data specifically for nX2 (or 2Xn) rectangles</a>
%H A331452 N. J. A. Sloane (in collaboration with Scott R. Shannon), <a href="/A331452/a331452.pdf">Art and Sequences</a>, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.
%H A331452 N. J. A. Sloane, <a href="https://vimeo.com/457349959">Conant's Gasket, Recamán Variations, the Enots Wolley Sequence, and Stained Glass Windows</a>, Experimental Math Seminar, Rutgers University, Sep 10 2020 (video of Zoom talk)
%e A331452 Triangle begins:
%e A331452      4;
%e A331452     16,   56;
%e A331452     46,  142,  340;
%e A331452    104,  296,  608,  1120;
%e A331452    214,  544, 1124,  1916,  3264;
%e A331452    380,  892, 1714,  2820,  4510,  6264;
%e A331452    648, 1436, 2678,  4304,  6888,  9360, 13968;
%e A331452   1028, 2136, 3764,  6024,  9132, 12308, 17758, 22904;
%e A331452   1562, 3066, 5412,  8126, 12396, 16592, 23604, 29374, 38748;
%e A331452   2256, 4272, 7118, 10792, 16226, 20896, 29488, 36812, 47050, 58256;
%e A331452   ...
%Y A331452 The first column is A306302, the main diagonal is A255011.
%Y A331452 The second column is A331766.
%Y A331452 See A333274 for the classification of vertices by valency.
%Y A331452 Cf. A288187, A331453, A331454, A333286, A333287, A333288.
%K A331452 nonn,tabl,nice
%O A331452 1,1
%A A331452 _Scott R. Shannon_ and _N. J. A. Sloane_, Jan 27 2020