
COMMENTS

If a(15) > 0 it is greater than 765.
Other than the 15gon, which contains a central 15gon by its construction, see A342222, the first kgon to contain a 15sided cell is the 399gon, but it does not contain a 13gon or a 14gon. The 647gon contains cells with total sides 312, 14, 15, but it does not contain a 13gon.
The 561gon contains cells with total sides 314, 16 but it does not contain a 15gon.
Other than the 17gon itself no kgon is currently known that contains a 17sided cell.
The first kgon to contain an 18sided cell is the 231gon, but it does not contain cells with total sides 13, 1517. Curiously the other known kgons to contain 18sided cells are 245, 469, 628, 708, and like the 231gon, all of these are also missing the 13, 1517 sided cells.
See the crossreferences for images of the kgons.


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 kgon is shown as '...'.
.
n  k  number of msided 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
