A355841 Irregular table read by rows: T(n,k) is the number of k-sided polygons formed, for k>=3, in a square when straight line segments connect the n+1 points along each edge that divide it into n equal parts to the n+1 points on the edge on the opposite side of the square.

4, 40, 128, 44, 12, 320, 152, 24, 616, 512, 84, 28, 1240, 744, 120, 40, 1936, 1928, 372, 136, 8, 3288, 2656, 616, 160, 4960, 4500, 1020, 332, 48, 7224, 6472, 1424, 392, 16, 9760, 11064, 2564, 824, 72, 16, 14144, 12424, 2696, 856, 32, 18312, 20604, 5308, 1468, 328, 16, 24384, 25392, 5968, 1584, 160, 8

Offset: 1

Author

Scott R. Shannon, Jul 18 2022

Keywords

• nonn
• tabf

Comments

This sequence is similar to A355801 but here the corner vertices of the square are also connected to points on the opposite edge.

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 A355838 for more images of the square.

Examples

			The table begins:
4;
40;
128,   44,    12;
320,   152,   24;
616,   512,   84,    28;
1240,  744,   120,   40;
1936,  1928,  372,   136,  8;
3288,  2656,  616,   160;
4960,  4500,  1020,  332,  48;
7224,  6472,  1424,  392,  16;
9760,  11064, 2564,  824,  72,  16;
14144, 12424, 2696,  856,  32;
18312, 20604, 5308,  1468, 328, 16;
24384, 25392, 5968,  1584, 160, 8;
31816, 32768, 7564,  2652, 240, 16;
40456, 42240, 10384, 3064, 248, 24;
49384, 59152, 15068, 4680, 704, 64;
.
.
		

Links

• Scott R. Shannon, Image for n = 10.

Crossrefs

Cf. A355838 (regions), A355839 (vertices), A355840 (edges), A355801 (without corner vertices), A290131, A331452, A335678.