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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. 1
4, 6, 7, 9, 15, 17, 35, 41, 71, 102, 202, 211 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,1

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.

LINKS

Table of n, a(n) for n=3..14.

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.

EXAMPLE

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

CROSSREFS

Cf. A342222, A342236, A007678, A331450.

Sequence in context: A005621 A140468 A342236 * A010418 A243193 A087790

Adjacent sequences: A349546 A349547 A349548 * A349550 A349551 A349552

KEYWORD

nonn,more,hard

AUTHOR

Scott R. Shannon, Nov 21 2021

STATUS

approved

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Last modified January 29 00:02 EST 2023. Contains 359905 sequences. (Running on oeis4.)