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

1,1

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.

Links

Table of n, a(n) for n=1..58.

Scott R. Shannon, Image for n = 10.

Example

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;
.
.

Crossrefs

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

Sequence in context: A091104 A069539 A063997 * A187378 A199833 A248964

Adjacent sequences: A355838 A355839 A355840 * A355842 A355843 A355844

Keyword

nonn,tabf

Author

Scott R. Shannon, Jul 18 2022

Status

approved