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.

1, 12, 75, 24, 3, 258, 132, 6, 621, 525, 33, 19, 1308, 1272, 144, 24, 2505, 2628, 345, 61, 4434, 4734, 984, 102, 12, 6, 7365, 7992, 1347, 243, 30, 9, 11556, 12552, 2412, 366, 48, 17073, 19266, 3969, 804, 60, 3, 0, 3, 0, 1, 24786, 27672, 6954, 1206, 186, 34611, 39066, 9099, 1768, 198, 27

Offset: 1

Author

Scott R. Shannon, Jul 18 2021

Keywords

• nonn
• tabf

Comments

See A344279 for other images of the polygons.

Examples

			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.
The table begins:
       1;
      12;
      75,     24,      3;
     258,    132,      6;
     621,    525,     33,    19;
    1308,   1272,    144,    24;
    2505,   2628,    345,    61;
    4434,   4734,    984,   102,   12,   6;
    7365,   7992,   1347,   243,   30,   9;
   11556,  12552,   2412,   366,   48;
   17073,  19266,   3969,   804,   60,   3,  0,  3, 0, 1;
   24786,  27672,   6954,  1206,  186;
   34611,  39066,   9099,  1768,  198,  27;
   47028,  53688,  15318,  2676,  288,  24;
   63039,  72210,  18513,  3708,  396,  75,  0,  6;
   82746,  93570,  24930,  4536,  498,  54, 18;
  106536, 121080,  32988,  6622,  678, 117,  6,  3;
  134520, 155748,  46326,  9456, 1266, 102, 12;
  167895, 196179,  55527, 11410, 1638, 156, 12,  3;
  207294, 243294,  74796, 15396, 2106, 276, 42,  6;
  254034, 297069,  87648, 17715, 2388, 363, 18,  3;
  308022, 360228, 108264, 21858, 3090, 282, 42, 18;
  370818, 433902, 132651, 28210, 4311, 486, 42,  9;
  440952, 520044, 168156, 36228, 5484, 720, 78;
  521031, 614526, 189297, 39541, 5790, 780, 60, 15;
  612990, 723228, 232980, 49278, 8004, 822, 96;
		

Links

• Scott R. Shannon, Image of the k-gons for n=3.

Crossrefs

Cf. A344279 (number of polygons), A344657 (number of vertices), A344896 (number of edges), A343755 (number of regions), A092867 (number polygons inside the triangle).

Formula

Sum of row(n) = A344279(n) = A344896(n) - A344657(n) + 1.