A355801 Irregular table read by rows: T(n,k) is the number of k-sided polygons, for k>=3, in a square when straight line segments connect the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on the edge on the opposite side of the square.

0, 1, 0, 4, 12, 12, 56, 32, 16, 156, 124, 24, 8, 0, 4, 384, 228, 72, 28, 716, 648, 144, 68, 8, 4, 1312, 1144, 240, 112, 8, 2244, 1912, 528, 256, 3528, 3072, 696, 360, 16, 5012, 5536, 1296, 524, 48, 28, 7696, 6596, 1960, 572, 16, 10340, 11448, 2968, 1028, 160, 24, 14520, 14428, 3872, 1156, 104, 8

Offset: 1

Author

Scott R. Shannon, Jul 17 2022

Keywords

• nonn
• tabf
• look

Comments

Up to n = 50 the maximum sided k-gon created is the 8-gon. It is plausible this is the maximum sided k-gon for all n, although this is unknown.

See A355798 for more images of the square.

The keyword "look" is for the n = 10 image. - N. J. A. Sloane, Jul 21 2022

Examples

			The table begins:
0,     1;
0,     4;
12,    12;
56,    32,    16;
156,   124,   24,   8,    0,   4;
384,   228,   72,   28;
716,   648,   144,  68,   8,   4;
1312,  1144,  240,  112,  8;
2244,  1912,  528,  256;
3528,  3072,  696,  360,  16;
5012,  5536,  1296, 524,  48,  28;
7696,  6596,  1960, 572,  16;
10340, 11448, 2968, 1028, 160, 24;
14520, 14428, 3872, 1156, 104, 8;
19588, 19156, 5296, 2052, 160, 8;
25392, 26112, 7160, 2152, 208, 24;
31820, 37244, 9936, 3240, 488, 64;
.
.
		

Links

• Scott R. Shannon, Image for n = 10.

Crossrefs

Cf. A355798 (regions), A355799 (vertices), A355800 (edges), A355801 (k-gons), A255011 (all vertices), A290131, A331452, A335678.