A344938 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

3,2

COMMENTS

See A344857 for examples and images of the polygons.

LINKS

Table of n, a(n) for n=3..69.

FORMULA

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

EXAMPLE

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:

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).

Sequence in context: A323085 A097548 A218047 * A261711 A128235 A357116

Adjacent sequences: A344935 A344936 A344937 * A344939 A344940 A344941

KEYWORD

nonn,tabf

AUTHOR

Scott R. Shannon, Jun 03 2021

STATUS

approved