A331450 -id:A331450 - cp's OEIS frontend

A349784 Maximum number of sides in any cell in a regular n-gon with all diagonals drawn (cf. A007678), excluding the central n-sided cell for odd values of n.

3, 3, 4, 5, 4, 6, 5, 6, 4, 8, 5, 7, 6, 8, 7, 8, 8, 7, 8, 8, 6, 8, 8, 8, 8, 11, 6, 10, 7, 8, 8, 9, 8, 10, 8, 8, 12, 10, 8, 14, 10, 9, 8, 10, 10, 10, 10, 12, 8, 12, 10, 10, 10, 10, 10, 12, 9, 12, 10, 12, 10, 12, 9, 10, 10, 10, 10, 11, 10, 12, 10, 12, 12, 10, 10, 10, 10, 10, 10, 10, 10, 12, 10

Offset: 4

Scott R. Shannon and N. J. A. Sloane, Nov 30 2021

nonn

As a regular n-gon with an odd number of sides always creates an n-sided cell at its center when all its diagonals are drawn, see A342222, this n-sided cell is not considered for odd n.
Although the behavior of the sequence is unknown as n -> infinity, the data up to n = 765 implies the sequence is possibly bounded. In the range studied the 14-gon is the predominant maximum-sided cell for n > 300.
No n-gon is currently known that produces a cell with 17 sides or 19 sides and above, other than the corresponding central n-sided cell for odd values of n.
See A342222 and A342236 for images of the n-gons.

			a(4) = 3 as a regular 4-gon (square) creates four 3-gons (triangles) when all its diagonals are drawn.
a(5) = 3 as a regular 5-gon (pentagon) creates ten 3-gons when all its diagonals are drawn. Also created is a central 5-gon but this cell is not considered.
a(6) = 4 as a regular 6-gon (hexagon) creates eighteen 3-gons and six 4-gons when all its diagonals are drawn.
a(7) = 5 as a regular 7-gon (heptagon) creates thirty-five 3-gons, seven 4-gons and seven 5-gons when all its diagonals are drawn. Also created is a central 7-gon but this cell is not considered.

Scott R. Shannon, Table of n, a(n) for n = 4..765
Scott R. Shannon, Image showing a close-up of the 18-sided cell in the 708-gon. Zoom in to see the vertices marked as white dots around the 18-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 1cm away from the inner vertex then the size of the entire 708-gon image would be slightly over 12.9 km in diameter.

Cf. A341729, A342236, A007678, A331450.

A344907 Number of polygon edges formed when every pair of vertices of a regular (2n-1)-gon are joined by an infinite line.

Original entry on oeis.org

0, 3, 30, 189, 684, 1815, 3978, 7665, 13464, 22059, 34230, 50853, 72900, 101439, 137634, 182745, 238128, 305235, 385614, 480909, 592860, 723303, 874170, 1047489, 1245384, 1470075, 1723878, 2009205, 2328564, 2684559, 3079890, 3517353, 3999840, 4530339, 5111934, 5747805, 6441228, 7195575

Offset: 1

Scott R. Shannon, Jun 02 2021

This sequences gives the number of polygon edges formed when connecting every pair of vertices of a regular polygon, with an odd number of vertices, by an infinite line.
A bisection of A344899. - N. J. A. Sloane, Sep 12 2021
See A344857 for other examples and images of the polygons.

			a(3) = 30 as the five connected vertices form a pentagon with fives lines along the pentagon's edges, fifteen lines inside forming eleven polygons, and ten lines outside forming another five triangles. In all these sixteen polygons form thirty edges. Twenty infinite edges between the outer unbounded regions are also formed.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A344899 (number of edges for all n-gons), A344866 (number of polygon), A146212, A344857, A344311, A007678, A331450, A344938.

A371274 Irregular table read by rows: T(n,k) is the number of k-sided regions, k>=2, formed when n equally spaced points are placed around a circle and all pairs of points are joined by an interior arc whose radius equals the circle's radius.

Original entry on oeis.org

1, 3, 3, 4, 0, 1, 10, 10, 5, 1, 12, 6, 14, 56, 21, 0, 7, 1, 8, 48, 32, 0, 0, 0, 1, 27, 144, 54, 27, 18, 10, 160, 70, 0, 30, 0, 0, 0, 1, 22, 253, 330, 11, 33, 0, 0, 0, 0, 1, 12, 276, 204, 0, 24, 26, 624, 403, 130, 104, 0, 0, 0, 0, 0, 0, 1, 14, 630, 448, 112, 70, 14, 14, 0, 0, 0, 0, 0, 1, 45, 960, 915, 165, 165

Offset: 2

Scott R. Shannon, Mar 17 2024

See A371253 and A371254 for images.

			The table begins:
1;
3, 3;
4, 0, 1;
10, 10, 5, 1;
12, 6;
14, 56, 21, 0, 7, 1;
8, 48, 32, 0, 0, 0, 1;
27, 144, 54, 27, 18;
10, 160, 70, 0, 30, 0, 0, 0, 1;
22, 253, 330, 11, 33, 0, 0, 0, 0, 1;
12, 276, 204, 0, 24;
26, 624, 403, 130, 104, 0, 0, 0, 0, 0, 0, 1;
14, 630, 448, 112, 70, 14, 14, 0, 0, 0, 0, 0, 1;
45, 960, 915, 165, 165;
16, 1136, 704, 272, 192, 0, 16, 0, 0, 0, 0, 0, 0, 0, 1;
34, 1581, 1870, 238, 272, 17, 34, 0, 0, 0, 0, 0, 0, 0, 0, 1;
18, 1656, 1386, 270, 288, 0, 18;
38, 2622, 2546, 646, 513, 38, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
20, 2680, 2420, 820, 380, 20, 60, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
63, 3297, 4725, 1050, 315, 42, 105;
22, 3696, 4136, 1342, 484, 22, 66, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

Cf. A371253 (regions), A371254 (vertices), A371255 (edges), A331450, A359009, A359061, A359258.

Sum of row n = A371253(n).

A341734 a(n) = A007678(2n)/(2n).

Original entry on oeis.org

0, 1, 4, 10, 22, 37, 68, 106, 137, 225, 310, 376, 538, 685, 716, 1058, 1288, 1471, 1842, 2170, 2327, 2941, 3388, 3734, 4412, 4993, 5444, 6306, 7042, 7391, 8680, 9586, 10289, 11585, 12682, 13628, 15078, 16381, 17440, 19210, 20740, 21899, 24038, 25810, 27245, 29613, 31648, 33418, 35992, 38305

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 07 2021

nonn

This is the number of cells in a 1/(2*n)-th sector of a regular (2*n)-gon with all diagonals drawn. See Rubinstein's illustrations in A007678.

			If we divide a regular hexagon with all diagonals drawn into 6 sectors (or pizza slices), each sector contains three triangles and one quadrilateral (cf. A331450), so a(3) = A007678(6)/6 = 24/6 = 4.

Scott R. Shannon, Table of n, a(n) for n = 1..71

Cf. A007678, A331450, A341735.

Row sums of triangle in A342268.

A350501 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of vertices in a regular n-gon after k generations of mitosis.

Original entry on oeis.org

3, 3, 4, 3, 5, 5, 3, 5, 10, 6, 3, 5, 15, 19, 7, 3, 5, 20, 25, 42, 8, 3, 5, 25, 25, 119, 57, 9, 3, 5, 30, 25, 231, 81, 135, 10, 3, 5, 35, 25, 378, 81, 504, 171, 11, 3, 5, 40, 25, 560, 81, 1017, 311, 341, 12, 3, 5, 45, 25, 777, 81, 1620, 361, 1309, 313, 13

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Jan 01 2022

See A350000 for further details and images of the n-gons.

			The table begins:
.
      |               Number of vertices after k generations
  n\k |  0,    1,     2,     3,     4,      5,      6,      7,      8,      9, ...
----------------------------------------------------------------------------------
   3  |  3,    3,     3,     3,     3,      3,      3,      3,      3,      3, ...
   4  |  4,    5,     5,     5,     5,      5,      5,      5,      5,      5, ...
   5  |  5,   10,    15,    20,    25,     30,     35,     40,     45,     50, ...
   6  |  6,   19,    25,    25,    25,     25,     25,     25,     25,     25, ...
   7  |  7,   42,   119,   231,   378,    560,    777,   1029,   1316,   1638, ...
   8  |  8,   57,    81,    81,    81,     81,     81,     81,     81,     81, ...
   9  |  9,  135,   504,  1017,  1620,   2313,   3096,   3969,   4932,   5985, ...
  10  | 10,  171,   311,   361,   411,    461,    511,    561,    611,    661, ...
  11  | 11,  341,  1309,  2629,  4169,   5929,   7909,  10109,  12529,   1516, ...
  12  | 12,  313,   481,   481,   481,    481,    481,    481,    481,    481, ...
  13  | 13,  728,  3601,  8125, 13624,  20098,  27547,  35971,  45370,  55744, ...
  14  | 14,  771,  1639,  2129,  2619,   3109,   3599,   4089,   4579,   5069, ...
  15  | 15, 1380,  5985, 13125, 22185,  32970,  45480,  59715,  75675,  93360, ...
  16  | 16, 1393,  3329,  4257,  4897,   5537,   6177,   6817,   7457,   8097, ...
  17  | 17, 2397, 12070, 28628, 50558,  77758, 110228, 147968, 190978, 239258, ...
  18  | 18, 1855,  4033,  5815,  7363,   8713,  10063,  11413,  12763,  14113, ...
  19  | 19, 3895, 19418, 44992, 77786, 117800, 165034, 219488, 281162, 350056, ...
  20  | 20, 3861, 11261, 16641, 20741,  24841,  28941,  33041,  37141,  41241, ...
  21  | 21, 6006, 26019, 55734, 92484, 136269, 187089, 244944, 309834, 381759, ...
  22  | 22, 5963, 18107, 27413, 34343,  41273,  48203,  55133,  62063,  68993, ...
.

Scott R. Shannon, Illustration of T(10,1).
Scott R. Shannon, Illustration of T(10,2).
Scott R. Shannon, Illustration of T(10,3).
Scott R. Shannon, Illustration of T(11,1).
Scott R. Shannon, Illustration of T(11,2).
Scott R. Shannon, Illustration of T(11,3).

Cf. A350000 (n-gons), A350502 (edges), A007569 (column 1), A349967 (column 2), A331450, A349968.

A342268 Irregular triangle read by rows: Take a regular (2n)-sided polygon (n>=2) with all diagonals drawn, as in A007678. Then T(n,k) = (1/(2n))*(number of k-sided polygons in that figure) for k = 3, 4, ..., max_k.

Original entry on oeis.org

1, 3, 1, 7, 3, 12, 9, 1, 23, 14, 34, 27, 7, 53, 42, 8, 3, 78, 53, 4, 1, 1, 110, 79, 29, 6, 0, 1, 136, 130, 37, 3, 2, 3, 184, 154, 35, 3, 184, 154, 35, 3, 297, 273, 76, 34, 4, 1, 389, 264, 48, 15, 449, 403, 153, 46, 7, 547, 497, 163, 69, 9, 3, 679, 519, 207, 59, 5, 2, 759, 717, 268, 71, 22, 5

Offset: 2

Scott R. Shannon and N. J. A. Sloane, Mar 07 2021

This is a version of A331450: take the even-indexed rows and divide by the number of vertices. That is, we only consider one sector (or pizza slice).

			Triangle begins:
1;
3, 1;
7, 3;
12, 9, 1;
23, 14;
34, 27, 7;
53, 42, 8, 3;
78, 53, 4, 1, 1;
110, 79, 29, 6, 0, 1;
136, 130, 37, 3, 2, 2;
184, 154, 35, 3;
242, 195, 81, 15, 4, 1;
297, 273, 76, 34, 4, 1;
389, 264, 48, 15;
449, 403, 153, 46, 7;
547, 497, 163, 69, 9, 3;
679, 519, 207, 59, 5, 2;
759, 717, 268, 71, 22, 5;
900, 819, 329, 100, 16, 5, 0, 0, 0, 1;
1079, 885, 271, 82, 9, 1;
...

Scott R. Shannon, Rows 2 through 70.

Cf. A007678.

Row sums give A341734.

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.

Original entry on oeis.org

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

Offset: 3

Scott R. Shannon, Nov 21 2021

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.

			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

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.

Cf. A342222, A342236, A007678, A331450.

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

A350502 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of edges in a regular n-gon after k generations of mitosis.

Original entry on oeis.org

3, 3, 4, 3, 8, 5, 3, 8, 20, 6, 3, 8, 35, 42, 7, 3, 8, 50, 66, 91, 8, 3, 8, 65, 66, 308, 136, 9, 3, 8, 80, 66, 630, 232, 288, 10, 3, 8, 95, 66, 1057, 232, 1305, 390, 11, 3, 8, 110, 66, 1589, 232, 2808, 900, 715, 12, 3, 8, 125, 66, 2226, 232, 4581, 1050, 3399, 756, 13

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Jan 01 2022

See A350000 for further details and images of the n-gons.

			The table begins:
.
      |               Number of edges after k generations
  n\k |  0,     1,     2,      3,      4,      5,      6,      7,      8, ...
----------------------------------------------------------------------------------
   3  |  3,     3,     3,      3,      3,      3,      3,      3,      3, ...
   4  |  4,     8,     8,      8,      8,      8,      8,      8,      8, ...
   5  |  5,    20,    35,     50,     65,     80,     95,    110,    125, ...
   6  |  6,    42,    66,     66,     66,     66,     66,     66,     66, ...
   7  |  7,    91,   308,    630,   1057,   1589,   2226,   2968,   3815, ...
   8  |  8,   136,   232,    232,    232,    232,    232,    232,    232, ...
   9  |  9,   288,  1305,   2808,   4581,   6624,   8937,  11520,  14373, ...
  10  | 10,   390,   900,   1050,   1200,   1350,   1500,   1650,   1800, ...
  11  | 11,   715,  3399,   7271,  11803,  16995,  22847,  29359,  36531, ...
  12  | 12,   756,  1428,   1428,   1428,   1428,   1428,   1428,   1428, ...
  13  | 13,  1508,  9061,  22243,  38350,  57382,  79339, 104221, 132028, ...
  14  | 14,  1722,  4704,   6174,   7644,   9114,  10584,  12054,  13524, ...
  15  | 15,  2835, 15345,  35880,  62370,  94035, 130875, 172890, 220080, ...
  16  | 16,  3088,  9424,  12496,  14416,  16336,  18256,  20176,  22096, ...
  17  | 17,  4896, 30294,  77758, 141440, 220932, 316234, 427346, 554268, ...
  18  | 18,  4320, 11376,  16686,  21528,  25578,  29628,  33678,  37728, ...
  19  | 19,  7923, 48773, 122607, 218101, 335255, 474069, 634543, 816677, ...
  20  | 20,  8360, 30840,  48260,  60560,  72860,  85160,  97460, 109760, ...
  21  | 21, 12180, 66738, 153069, 260505, 389046, 538692, 709443, 901299, ...
  22  | 22, 12782, 49148,  79442, 100232, 121022, 141812, 162602, 183392, ...
.

Cf. A350000 (n-gons), A350501 (vertices), A135565 (column 1), A349968 (column 2), A331450, A349967.

A340650 Irregular table read by rows: row n gives the number of 7-gon to k-gon contacts, with k>=7, for a regular n-gon with all diagonals drawn, with n>=19.

Original entry on oeis.org

19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 52, 0, 0, 0, 0, 0, 0, 0, 118, 0, 61, 0, 63, 0, 0, 0, 0, 0, 138, 70, 0, 72, 73, 148, 0, 0, 0, 0, 158, 80, 0, 82, 83, 0, 0, 172, 0, 0, 178, 540, 0, 0, 93, 0, 93, 282, 95, 96, 194, 294, 0, 198, 99

Offset: 19

Scott R. Shannon, Jan 14 2021

For n=3 to n=18 there are no n-gons that have 7-gon to k-gon contacts, where k>=7, so the table starts at n=19.
See A333654 for the number of 3-gon to k-gon contacts, with k>=3.
See A335614 for the number of 4-gon to k-gon contacts, with k>=4.
See A335646 for the number of 5-gon to k-gon contacts, with k>=5.
See A337330 for the number of 6-gon to k-gon contacts, with k>=6.
See A007678 for the number of regions and images of other n-gons.

			a(19) = 19, a(20-51) = 0, a(52) = 52, a(53-58) = 0.
The table from a(59) begins:
0, 118,
9;
61;
0;
63;
0;
0;
0;
0;
0;
138;
70;
0;
72;
73;
148;
0;
0;
0;
0;
158;
80;
0;
82;
83;
0;
0;
172;
0;
0;
178;
540;
0;
0;
93, 0, 93;
282;
95;
96;
194;
294;
0, 198, 99;
0;

Scott R. Shannon, Image for n = 19. This is the first n-gon with 7-gon to 7-gon contacts.
Scott R. Shannon, Image for n = 59. This is the first n-gon with 7-gon to 8-gon contacts.

Cf. A333654 (3-gon contacts), A335614 (4-gon contacts), A335646 (5-gon contacts), A337330 (6-gon contacts), A007678, A135565, A007569, A062361, A331450, A331451.

A342269 Take a regular (2*n+1)-gon with all diagonals drawn; a(n) = number of edges in next-to-largest cell.

Original entry on oeis.org

3, 5, 6, 6, 8, 7, 8, 8, 7, 8, 8, 8, 11, 10, 8, 9, 10, 8, 10, 14, 9, 10, 10, 12, 12, 10, 10, 12, 12, 12, 12, 10, 10, 11, 12, 12, 10, 10, 10, 10, 12, 10, 10, 12, 13, 12, 12, 11, 12, 12, 12, 12, 12, 10, 10, 12, 14, 14, 12, 12, 12, 10, 12, 12, 14, 12, 10, 12, 12

Offset: 2

Scott R. Shannon and N. J. A. Sloane, Mar 07 2021

The largest cell has 2*n+1 sides, this is the runner-up.
It can be read off the rows of A331450.
It would be nice to know how fast this sequence grows. Is it unbounded?

Scott R. Shannon, Image for a(2) = 3. The 5-gon has a next-to-largest cell with 3 edges.
Scott R. Shannon, Image for a(3) = 5. The 7-gon has a next-to-largest cell with 5 edges.
Scott R. Shannon, Image for a(4) = 6. The 9-gon has a next-to-largest cell with 6 edges.
Scott R. Shannon, Image for a(7) = 7. The 15-gon has a next-to-largest cell with 7 edges.
Scott R. Shannon, Image for a(6) = 8. The 13-gon has a next-to-largest cell with 8 edges.
Scott R. Shannon, Image for a(17) = 9. The 35-gon has a next-to-largest cell with 9 edges.
Scott R. Shannon, Image for a(15) = 10. The 31-gon has a next-to-largest cell with 10 edges.
Scott R. Shannon, Image for a(14) = 11. The 29-gon has a next-to-largest cell with 11 edges.
Scott R. Shannon, Image for a(25) = 12. The 51-gon has a next-to-largest cell with 12 edges.

Cf. A331450, A342236, A341735, A341729, A342222.

cp's OEIS Frontend

A349784 Maximum number of sides in any cell in a regular n-gon with all diagonals drawn (cf. A007678), excluding the central n-sided cell for odd values of n.

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A344907 Number of polygon edges formed when every pair of vertices of a regular (2n-1)-gon are joined by an infinite line.

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A371274 Irregular table read by rows: T(n,k) is the number of k-sided regions, k>=2, formed when n equally spaced points are placed around a circle and all pairs of points are joined by an interior arc whose radius equals the circle's radius.

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A341734 a(n) = A007678(2*n)/(2*n).

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A350501 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of vertices in a regular n-gon after k generations of mitosis.

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A342268 Irregular triangle read by rows: Take a regular (2*n)-sided polygon (n>=2) with all diagonals drawn, as in A007678. Then T(n,k) = (1/(2*n))*(number of k-sided polygons in that figure) for k = 3, 4, ..., max_k.

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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.

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A350502 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of edges in a regular n-gon after k generations of mitosis.

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A340650 Irregular table read by rows: row n gives the number of 7-gon to k-gon contacts, with k>=7, for a regular n-gon with all diagonals drawn, with n>=19.

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A342269 Take a regular (2*n+1)-gon with all diagonals drawn; a(n) = number of edges in next-to-largest cell.

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A341734 a(n) = A007678(2n)/(2n).

A342268 Irregular triangle read by rows: Take a regular (2n)-sided polygon (n>=2) with all diagonals drawn, as in A007678. Then T(n,k) = (1/(2n))*(number of k-sided polygons in that figure) for k = 3, 4, ..., max_k.