A356119 Irregular table read by rows: T(n,k) is the number of k-sided polygons formed, for k>=3, in an equilateral triangle when straight line segments connect the three corner vertices to the points dividing the sides into n equal parts.

1, 6, 12, 3, 3, 1, 24, 6, 36, 9, 9, 7, 48, 24, 6, 72, 21, 15, 19, 84, 48, 12, 6, 108, 51, 33, 25, 132, 78, 18, 18, 168, 69, 51, 43, 180, 120, 48, 18, 216, 135, 57, 61, 252, 156, 66, 36, 294, 159, 105, 67, 312, 234, 84, 48, 372, 225, 117, 103, 408, 264, 138, 60, 456, 291, 159, 121, 486, 372, 138, 84

Offset: 1

Author

Scott R. Shannon, Jul 27 2022

Keywords

• nonn
• tabf

Comments

The maximum sided k-gon up to n = 250 is the 6-gon; it is likely this is the maximum sided k-gon for all n.

See A092098 for more images of the triangle.

Examples

			The table begins:
1;
6;
12,  3,   3,   1;
24,  6;
36,  9,   9,   7;
48,  24,  6;
72,  21,  15,  19;
84,  48,  12,  6;
108, 51,  33,  25;
132, 78,  18,  18;
168, 69,  51,  43;
180, 120, 48,  18;
216, 135, 57,  61;
252, 156, 66,  36;
294, 159, 105, 67;
312, 234, 84,  48;
372, 225, 117, 103;
408, 264, 138, 60;
456, 291, 159, 121;
486, 372, 138, 84;
.
.
See the attached text file for more examples.
		

Links

• Scott R. Shannon, Table for n = 1..250.
• Scott R. Shannon, Image for n = 20.

Crossrefs

Cf. A092098 (regions), vertices (A331782), edges (A332376), A335413.

Formula

Sum of row(n) = A092098(n).