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A335701 Irregular triangle read by rows: consider the structure formed by drawing the lines connecting any two of the 2*(n+2) perimeter points of an (n+1) X 3 rectangular grid of points (or equally, an n X 2 grid of squares); row n gives number of cells with k sides, for k >= 3.

%I A335701 #54 May 21 2021 07:03:58
%S A335701 14,2,48,8,102,36,4,192,92,12,326,194,24,524,336,28,4,802,554,80,1192,
%T A335701 812,128,4,1634,1314,112,0,4,2,2296,1756,200,20,3074,2508,236,22,4052,
%U A335701 3252,356,28,5246,4348,472,28,6740,5464,652,28,8398,7054,656,74,10440,8760,940,52,12770,11050,1040,58,15512,13324,1300,60,4,18782,16162,1600,70,22384,19256,1948,104
%N A335701 Irregular triangle read by rows: consider the structure formed by drawing the lines connecting any two of the 2*(n+2) perimeter points of an (n+1) X 3 rectangular grid of points (or equally, an n X 2 grid of squares); row n gives number of cells with k sides, for k >= 3.
%C A335701 More than the usual number of terms are given, in order to include the first 20 rows and emphasize the fact that so far k is never more than 8.
%C A335701 These are the structures discussed in column 2 of the table in A331452. It is known that the structures discussed in column 1 of that table have cells with at most 4 sides, so an upper limit of 8 sides for the present sequence is certainly possible.
%C A335701 The maximum number of sides for n=19..106 is 6. - _Lars Blomberg_, Aug 27 2020
%H A335701 Lars Blomberg, <a href="/A335701/b335701.txt">Table of n, a(n) for n = 1..419</a> (the first 106 rows)
%H A335701 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 A335701 Scott R. Shannon, <a href="/A335701/a335701_12.png">Colored illustration for n=1</a>
%H A335701 Scott R. Shannon, <a href="/A335701/a335701_13.png">Colored illustration for n=2</a>
%H A335701 Scott R. Shannon, <a href="/A335701/a335701_14.png">Colored illustration for n=3</a>
%H A335701 Scott R. Shannon, <a href="/A335701/a335701_15.png">Colored illustration for n=4</a>
%H A335701 Scott R. Shannon, <a href="/A335701/a335701_16.png">Colored illustration for n=5</a>
%H A335701 Scott R. Shannon, <a href="/A335701/a335701_17.png">Colored illustration for n=6</a>
%H A335701 Scott R. Shannon, <a href="/A335701/a335701_1.png">Colored illustration for n=7</a>
%H A335701 Scott R. Shannon, <a href="/A335701/a335701.png">Colored illustration for n=8</a>
%H A335701 Scott R. Shannon, <a href="/A335701/a335701_18.png">Colored illustration for n=9</a>
%H A335701 Scott R. Shannon, <a href="/A331452/a331452_31.png">Colored illustration for n=10</a>
%H A335701 Scott R. Shannon, <a href="/A335701/a335701_5.png">Colored illustration for n=11</a>
%H A335701 Scott R. Shannon, <a href="/A335701/a335701_2.png">Colored illustration for n=12</a>
%H A335701 Scott R. Shannon, <a href="/A335701/a335701_3.png">Colored illustration for n=13</a>
%H A335701 Scott R. Shannon, <a href="/A335701/a335701_4.png">Colored illustration for n=14</a>
%H A335701 Scott R. Shannon, <a href="/A335701/a335701_6.png">Colored illustration for n=15</a>
%H A335701 Scott R. Shannon, <a href="/A335701/a335701_11.png">Colored illustration for n=16</a>
%H A335701 Scott R. Shannon, <a href="/A331452/a331452_12.png">Colored illustration for n=2</a>. Random distance-based coloring.
%H A335701 Scott R. Shannon, <a href="/A331452/a331452_13.png">Colored illustration for n=3</a>. Random distance-based coloring.
%H A335701 Scott R. Shannon, <a href="/A331452/a331452_14.png">Colored illustration for n=4</a>. Random distance-based coloring.
%H A335701 Scott R. Shannon, <a href="/A331452/a331452_15.png">Colored illustration for n=5</a>. Random distance-based coloring.
%H A335701 Scott R. Shannon, <a href="/A331452/a331452_16.png">Colored illustration for n=6</a>. Random distance-based coloring.
%H A335701 Scott R. Shannon, <a href="/A335701/a335701_19.png">Colored illustration for n=7</a>. Random distance-based coloring.
%H A335701 Scott R. Shannon, <a href="/A335701/a335701_20.png">Colored illustration for n=8</a>. Random distance-based coloring.
%H A335701 Scott R. Shannon, <a href="/A335701/a335701_21.png">Colored illustration for n=9</a>. Random distance-based coloring.
%H A335701 Scott R. Shannon, <a href="/A335701/a335701_22.png">Colored illustration for n=10</a>. Random distance-based coloring.
%e A335701 Triangle begins:
%e A335701 14, 2,
%e A335701 48, 8,
%e A335701 102, 36, 4,
%e A335701 192, 92, 12
%e A335701 326, 194, 24
%e A335701 524, 336, 28, 4
%e A335701 802, 554, 80,
%e A335701 1192, 812, 128, 4
%e A335701 1634, 1314, 112, 0, 4, 2
%e A335701 2296, 1756, 200, 20
%e A335701 3074, 2508, 236, 22
%e A335701 4052, 3252, 356, 28
%e A335701 5246, 4348, 472, 28
%e A335701 6740, 5464, 652, 28
%e A335701 8398, 7054, 656, 74
%e A335701 10440, 8760, 940, 52
%e A335701 12770, 11050, 1040, 58
%e A335701 15512, 13324, 1300, 60, 4
%e A335701 18782, 16162, 1600, 70
%e A335701 22384, 19256, 1948, 104
%e A335701 ...
%e A335701 The 1X2 structure (or 2X1 structure, as in the illustration) contains 14 triangles and 2 quadrilaterals, so row 1 is 14, 2.
%e A335701 The 3X2 structure contains 102 triangles, 36 quadrilaterals, and 4 pentagons, so row 3 is 102, 36, 4. The sum is 142 = A331766(3).
%Y A335701 Cf. A331452, A331766 (row sums), A331763, A331765.
%K A335701 nonn,tabf
%O A335701 1,1
%A A335701 _Scott R. Shannon_ and _N. J. A. Sloane_, Aug 09 2020