A344938 Irregular triangle read by rows: T(n,k) = number of k-sided polygons formed when every pair of vertices of a regular n-gon are joined by an infinite line, for k = 3, 4, ..., max_k.

1, 4, 15, 0, 1, 36, 6, 70, 21, 7, 0, 1, 112, 64, 189, 108, 36, 18, 0, 0, 1, 270, 220, 50, 407, 352, 110, 55, 0, 0, 0, 0, 1, 624, 528, 884, 689, 325, 91, 0, 26, 0, 0, 0, 0, 1, 1162, 1092, 266, 14, 1530, 1545, 480, 270, 45, 0, 0, 0, 0, 0, 0, 0, 1, 2080, 2032, 416, 80

Offset: 3

Author

Scott R. Shannon, Jun 03 2021

Keywords

• nonn
• tabf

Comments

See A344857 for examples and images of the polygons.

Examples

			A pentagon with all vertices connected forms 10 triangles inside the pentagon, 5 triangles outside the pentagon, giving 15 triangles in all, and 1 smaller pentagon inside the pentagon, so row 3 is [15,0,1].
The table begins:
1;
4;
15,0,1;
36,6;
70,21,7,0,1;
112,64;
189,108,36,18,0,0,1;
270,220,50;
407,352,110,55,0,0,0,0,1;
624,528;
884,689,325,91,0,26,0,0,0,0,1;
1162,1092,266,14;
1530,1545,480,270,45,0,0,0,0,0,0,0,1;
2080,2032,416,80;
2567,2754,1003,374,17,68,0,0,0,0,0,0,0,0,1;
3402,3366,180,18,18;
3952,4807,1672,475,95,76,0,19,0,0,0,0,0,0,0,0,1;
5380,5360,1580,240,0,20;
5943,7392,2583,1260,21,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
7590,9020,2310,132,132,66;
9430,9775,4508,1518,253,46,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
11304,12288,2280,144;
13025,14650,6250,2375,200,75,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
16042,16952,5954,728,260,52;
17064,22464,7884,2700,567,189,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
21616,24192,7056,2016,168,28;
23751,29319,11281,3828,348,319,0,87,29,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
29880,29010,4140,540;
30814,39370,15314,5177,341,496,0,62,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
37440,42624,14240,3008,544,64;
41481,49335,19305,7854,891,363,66,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
		

Crossrefs

Cf. A344857 (total number of polygons), A344899 (number of edges), A146212 (number of vertices), A344866, A344311, A007678, A331450 (number of k-gons inside the regular n-gon).

Formula

Sum of row(n) = A344857(n) = A344311(n) + A007678(n).