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A346446 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.

%I A346446 #13 Jul 21 2021 09:13:47
%S A346446 1,12,75,24,3,258,132,6,621,525,33,19,1308,1272,144,24,2505,2628,345,
%T A346446 61,4434,4734,984,102,12,6,7365,7992,1347,243,30,9,11556,12552,2412,
%U A346446 366,48,17073,19266,3969,804,60,3,0,3,0,1,24786,27672,6954,1206,186,34611,39066,9099,1768,198,27
%N A346446 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 A346446 See A344279 for other images of the polygons.
%H A346446 Scott R. Shannon, <a href="/A346446/a346446.gif">Image of the k-gons for n=3</a>.
%F A346446 Sum of row(n) = A344279(n) = A344896(n) - A344657(n) + 1.
%e A346446 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 A346446 The table begins:
%e A346446        1;
%e A346446       12;
%e A346446       75,     24,      3;
%e A346446      258,    132,      6;
%e A346446      621,    525,     33,    19;
%e A346446     1308,   1272,    144,    24;
%e A346446     2505,   2628,    345,    61;
%e A346446     4434,   4734,    984,   102,   12,   6;
%e A346446     7365,   7992,   1347,   243,   30,   9;
%e A346446    11556,  12552,   2412,   366,   48;
%e A346446    17073,  19266,   3969,   804,   60,   3,  0,  3, 0, 1;
%e A346446    24786,  27672,   6954,  1206,  186;
%e A346446    34611,  39066,   9099,  1768,  198,  27;
%e A346446    47028,  53688,  15318,  2676,  288,  24;
%e A346446    63039,  72210,  18513,  3708,  396,  75,  0,  6;
%e A346446    82746,  93570,  24930,  4536,  498,  54, 18;
%e A346446   106536, 121080,  32988,  6622,  678, 117,  6,  3;
%e A346446   134520, 155748,  46326,  9456, 1266, 102, 12;
%e A346446   167895, 196179,  55527, 11410, 1638, 156, 12,  3;
%e A346446   207294, 243294,  74796, 15396, 2106, 276, 42,  6;
%e A346446   254034, 297069,  87648, 17715, 2388, 363, 18,  3;
%e A346446   308022, 360228, 108264, 21858, 3090, 282, 42, 18;
%e A346446   370818, 433902, 132651, 28210, 4311, 486, 42,  9;
%e A346446   440952, 520044, 168156, 36228, 5484, 720, 78;
%e A346446   521031, 614526, 189297, 39541, 5790, 780, 60, 15;
%e A346446   612990, 723228, 232980, 49278, 8004, 822, 96;
%Y A346446 Cf. A344279 (number of polygons), A344657 (number of vertices), A344896 (number of edges), A343755 (number of regions), A092867 (number polygons inside the triangle).
%K A346446 nonn,tabf
%O A346446 1,2
%A A346446 _Scott R. Shannon_, Jul 18 2021