A349549 The smallest k such that a regular k-gon with all diagonals drawn contains cells with a total number of sides of 3 through n, or -1 if no such k exists.

4, 6, 7, 9, 15, 17, 35, 41, 71, 102, 202, 211, 843

Offset: 3

Author

Scott R. Shannon, Nov 21 2021

Keywords

• nonn
• more
• hard

Comments

If a(15) > 0 it is greater than 765.

Other than the 15-gon, which contains a central 15-gon by its construction, see A342222, the first k-gon to contain a 15-sided cell is the 399-gon, but it does not contain a 13-gon or a 14-gon. The 647-gon contains cells with total sides 3-12, 14, 15, but it does not contain a 13-gon.

The 561-gon contains cells with total sides 3-14, 16 but it does not contain a 15-gon.

Other than the 17-gon itself no k-gon is currently known that contains a 17-sided cell.

The first k-gon to contain an 18-sided cell is the 231-gon, but it does not contain cells with total sides 13, 15-17. Curiously the other known k-gons to contain 18-sided cells are 245, 469, 628, 708, and like the 231-gon, all of these are also missing the 13, 15-17 sided cells.

See the cross-references for images of the k-gons.

Examples

			The number of cells containing m sides, where 3 <= m <= n, is given below for the currently known values of n. For odd k the list of 0's leading up to the single central k-gon is shown as '...'.
.
  n  | k   | number of m-sided cells, 3 <= m <= n
-----------------------------------------------------
  3  | 4   | 4
  4  | 6   | 18, 6
  5  | 7   | 35, 7, 7, 0, 1
  6  | 9   | 90, 36, 18, 9, 0, 0, 1
  7  | 15  | 585, 600, 150, 105, 15, ..., 1
  8  | 17  | 1054, 901, 357, 136, 17, 34, ..., 1
  9  | 35  | 19705, 20475, 8190, 3640, 560, 315, 35, ..., 1
  10 | 41  | 39278, 37064, 16564, 7298, 1025, 656, 123, 41, ..., 1
  11 | 71  | 361319, 359118, 172246, 65604, 10934, 4118, 568, 71, 71, ..., 1
  12 | 102 | 1587732, 1547238, 699414, 222870, 41616, 9486, 306, 918, 102, 102
  13 | 202 | 24468260, 25271008, 11988296, 3828102, 777700, 171296, 16968, \
                                                              6060, 404, 404, 202
  14 | 211 | 28946246, 30389486, 14708177, 4895411, 1025882, 281896, 14981, \
                                                18568, 633, 422, 211, 211, ..., 1
  15 | 843 | 7465441086, 7927237329, 3927037101, 1250023161, 266472300, \
                 50115507, 5487930, 1534260, 44679, 95259, 843, 3372, 843, ..., 1
		

Links

• Scott R. Shannon, Image showing a close-up of the 14-sided cell in the 211-gon. Zoom in to see the vertices marked as white dots around the 14-gon, shown in violet. The bottom three vertices are extremely close together -- if the image were expanded so that the outer two of these vertices were 1 cm away from the inner vertex then the size of the entire 211-gon image would be slightly over 3.5 km in diameter.

Crossrefs

Cf. A342222, A342236, A007678, A331450.

Extensions

a(15) from Scott R. Shannon, May 28 2023