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%I #13 Jul 21 2021 09:13:47
%S 1,12,75,24,3,258,132,6,621,525,33,19,1308,1272,144,24,2505,2628,345,
%T 61,4434,4734,984,102,12,6,7365,7992,1347,243,30,9,11556,12552,2412,
%U 366,48,17073,19266,3969,804,60,3,0,3,0,1,24786,27672,6954,1206,186,34611,39066,9099,1768,198,27
%N Irregular triangle read by rows: T(n,k) = number of k-sided polygons formed when connecting infinite lines between all vertices and all points that divide the sides of an equilateral triangle into n equal parts, for k = 3, 4, ..., max_k.
%C See A344279 for other images of the polygons.
%H Scott R. Shannon, <a href="/A346446/a346446.gif">Image of the k-gons for n=3</a>.
%F Sum of row(n) = A344279(n) = A344896(n) - A344657(n) + 1.
%e Connecting infinite lines between an equilateral triangle's three vertices and the two points along each side that divide the sides into three equal parts forms seventy-five triangles, twenty-four quadrilaterals and three pentagons, so row 3 is [75,24,3]. See the linked image.
%e The table begins:
%e 1;
%e 12;
%e 75, 24, 3;
%e 258, 132, 6;
%e 621, 525, 33, 19;
%e 1308, 1272, 144, 24;
%e 2505, 2628, 345, 61;
%e 4434, 4734, 984, 102, 12, 6;
%e 7365, 7992, 1347, 243, 30, 9;
%e 11556, 12552, 2412, 366, 48;
%e 17073, 19266, 3969, 804, 60, 3, 0, 3, 0, 1;
%e 24786, 27672, 6954, 1206, 186;
%e 34611, 39066, 9099, 1768, 198, 27;
%e 47028, 53688, 15318, 2676, 288, 24;
%e 63039, 72210, 18513, 3708, 396, 75, 0, 6;
%e 82746, 93570, 24930, 4536, 498, 54, 18;
%e 106536, 121080, 32988, 6622, 678, 117, 6, 3;
%e 134520, 155748, 46326, 9456, 1266, 102, 12;
%e 167895, 196179, 55527, 11410, 1638, 156, 12, 3;
%e 207294, 243294, 74796, 15396, 2106, 276, 42, 6;
%e 254034, 297069, 87648, 17715, 2388, 363, 18, 3;
%e 308022, 360228, 108264, 21858, 3090, 282, 42, 18;
%e 370818, 433902, 132651, 28210, 4311, 486, 42, 9;
%e 440952, 520044, 168156, 36228, 5484, 720, 78;
%e 521031, 614526, 189297, 39541, 5790, 780, 60, 15;
%e 612990, 723228, 232980, 49278, 8004, 822, 96;
%Y Cf. A344279 (number of polygons), A344657 (number of vertices), A344896 (number of edges), A343755 (number of regions), A092867 (number polygons inside the triangle).
%K nonn,tabf
%O 1,2
%A _Scott R. Shannon_, Jul 18 2021
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