A331451 -id:A331451 - cp's OEIS frontend

A342236 a(n) is the smallest m such that a regular m-gon with all diagonals drawn contains a cell with n sides, as in A342222, but for odd m the central m-sided polygon is not considered. Otherwise a(n) = -1 if no such m exists.

4, 6, 7, 9, 15, 13, 35, 29, 29, 40, 93, 43, 399, 212

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Mar 06 2021

An m-gon with an odd number of sides contains a central cell with m sides by its construction, and it will be the m-gon with the fewest possible sides to do so. See A342222 for a proof. This sequence lists the smallest m-sided polygon to contain an n-sided cell where this central cell is not considered for odd m.
See A342222 for other images of the m-sided polygons.
a(17) is presently unknown, but if a(17) > 0 it is greater than 765.

Scott R. Shannon, Image for a(3) = 4.
Scott R. Shannon, Image for a(9) = 35.
Scott R. Shannon, Image for a(13) = 93.

Cf. A342222, A007678, A331450, A331451, A007569, A135565.

See also A341729 and A341730 for the maximum number of sides in any cell.

a(15)-a(16) added by Scott R. Shannon, Mar 15 2021
Minimum value for a(17) updated by Scott R. Shannon, Mar 21 2021
Minimum value for a(17) updated by Scott R. Shannon, Nov 30 2021

A333037 Table read by rows: T(n,k) = number of k-sided polygons in an equal-armed cross with arms of length n (see Comments in A331456 for definition) for k = 3,4,5,6,7.

Original entry on oeis.org

4, 0, 0, 0, 0, 84, 20, 0, 0, 0, 380, 180, 0, 8, 0, 1076, 764, 40, 20, 0, 2380, 2316, 64, 48, 0, 4716, 5188, 224, 52, 0, 8236, 10492, 360, 92, 0, 13620, 18772, 632, 108, 0, 21188, 31380, 864, 196, 0, 31596, 49228, 1376, 224, 8, 44980, 74268, 1920, 272, 8

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Mar 06 2020

See the links in A331456 for images of the crosses.

			Table begins:
4, 0, 0, 0, 0
84, 20, 0, 0, 0
380, 180, 0, 8, 0
1076, 764, 40, 20, 0
2380, 2316, 64, 48, 0
4716, 5188, 224, 52, 0
8236, 10492, 360, 92, 0
13620, 18772, 632, 108, 0
21188, 31380, 864, 196, 0
31596, 49228, 1376, 224, 8
44980, 74268, 1920, 272, 8
The row sums are A331456.

Cf. A331456 (regions), A333035 (vertices), A333036 (edges), A331451, A332723, A333025.

A341730 Maximum number of sides in any cell in a regular 2n-gon with all diagonals drawn (cf. A007678).

Original entry on oeis.org

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

Offset: 2

Scott R. Shannon and N. J. A. Sloane, Mar 06 2021

nonn

A bisection of A341729.
a(46) = 14, corresponding to a regular 92-gon, suggesting that this sequence may be unbounded (cf. A342222). (The Pfetsch-Ziegler web page discusses a similar question for polygons defined by grid points.) It would be nice to have a b-file.
For which values of n is a(n) odd (and why)?

M. E. Pfetsch and G. M. Ziegler, Large chambers in a lattice polygon, December 13, 2004;
M. E. Pfetsch and G. M. Ziegler, Large chambers in a lattice polygon [Local copy of webpage, pdf only]

Cf. A007678, A062361, A331450, A331451, A341729, A342222.

A333458 Irregular table read by rows: Take a diagonal-edged (or diamond-shaped) checkerboard with all diagonals drawn, as in A333434. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

Original entry on oeis.org

4, 84, 20, 692, 312, 72, 4, 2708, 2020, 400, 88, 0, 4, 7876, 5764, 1580, 264, 20, 4, 18220, 14868, 4960, 904, 112, 24, 36764, 30624, 11224, 2496, 304, 52, 66004, 55192, 18624, 3952, 472, 48, 111764, 97840, 40008, 9912, 1740, 244, 36, 175156, 158280, 63632, 15744, 2560, 156, 8, 16

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 22 2020

See A333434 for the definition and images of the checkerboard.

			Table begins:
4;
84, 20;
692, 312, 72, 4;
2708, 2020, 400, 88, 0, 4;
7876, 5764, 1580, 264, 20, 4;
18220, 14868, 4960, 904, 112, 24;
36764, 30624, 11224, 2496, 304, 52;
66004, 55192, 18624, 3952, 472, 48;
111764, 97840, 40008, 9912, 1740, 244, 36;
175156, 158280, 63632, 15744, 2560, 156, 8, 16;
...

Lars Blomberg, Table of n, a(n) for n = 1..201 (the first 27 rows)

Cf. A333434 (regions), A333459 (vertices), A333460 (edges), A331911, A331451, A333025, A333037.

a(32) and beyond from Lars Blomberg, Jun 03 2020

A342153 Irregular table read by rows: Take a vesica piscis with all diagonals drawn, as in A341877. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

Original entry on oeis.org

0, 4, 18, 6, 52, 28, 4, 120, 78, 34, 4, 252, 188, 56, 12, 470, 348, 184, 40, 4, 2, 808, 648, 300, 56, 8, 1282, 1118, 548, 138, 20, 4, 2036, 1772, 644, 156, 28, 8, 2878, 2804, 1252, 388, 96, 10, 4172, 4024, 1728, 468, 100, 28, 5752, 5682, 2600, 866, 162, 46, 7912, 7676, 3420, 1024, 196, 44, 16

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 02 2021

The terms are from numeric computation - no formula for a(n) is currently known.

See A341877 for images of the regions and A341878 for images of the vertices.

			A vesica piscis with 1 point dividing its edges, n = 2, contains 4 triangles and no other n-gons, so the second row is [4]. A vesica piscis with 3 points dividing its edges, n = 4, contains 52 triangles, 28 quadrilaterals, 4 pentagons and no other n-gons, so the fourth row is [52, 28, 4].
The table begins:
0;
4;
18,6;
52,28,4;
120,78,34,4;
252,188,56,12;
470,348,184,40,4,2;
808,648,300,56,8;
1282,1118,548,138,20,4;
2036,1772,644,156,28,8;
2878,2804,1252,388,96,10;
4172,4024,1728,468,100,28;
5752,5682,2600,866,162,46;
7912,7676,3420,1024,196,44,16;
10388,10354,4868,1548,352,60,6;
13496,13808,6016,1836,388,80,4,0,4;
17310,17590,8376,2672,564,122,16,2;
22012,22364,10160,3152,712,124,20,4;
27440,27956,13162,4432,964,172,24,2,4;
33784,34736,15588,4640,1096,120,28;

Scott R. Shannon, Image of the k-gons for n=14.
Wikipedia, Vesica piscis.

Cf. A341877 (regions), A342152 (edges), A341878 (vertices), A331451, A331911, A340614, A340688.

A359119 Irregular table read by rows: T(n,k) is the number of k-gons, k>=2, in the Farey Ring graph FR(n) defined in A359116.

Original entry on oeis.org

2, 4, 4, 6, 18, 6, 10, 124, 76, 32, 8, 12, 244, 196, 78, 14, 4, 18, 1184, 1296, 534, 118, 28, 2, 22, 2632, 3180, 1244, 330, 58, 2, 28, 7244, 8628, 3594, 962, 190, 38, 32, 12626, 14922, 6378, 1836, 330, 36, 4, 42, 39060, 45656, 20152, 6082, 1252, 132, 28, 2, 46, 56980, 66088, 29454, 8916, 1840, 244, 26, 6

Offset: 2

Scott R. Shannon and N. J. A. Sloane, Dec 17 2022

See A359116 and A359117 for further images of the graphs.

			The table begins:
2;
4, 4;
6, 18, 6;
10, 124, 76, 32, 8;
12, 244, 196, 78, 14, 4;
18, 1184, 1296, 534, 118, 28, 2;
22, 2632, 3180, 1244, 330, 58, 2;
28, 7244, 8628, 3594, 962, 190, 38;
32, 12626, 14922, 6378, 1836, 330, 36, 4;
42, 39060, 45656, 20152, 6082, 1252, 132, 28, 2;
46, 56980, 66088, 29454, 8916, 1840, 244, 26, 6;
58, 148058, 170352, 76834, 23936, 4990, 766, 136, 12, 0, 2;
64, 221744, 253808, 115806, 35922, 7428, 1286, 136, 14;
72, 359676, 408252, 188438, 57828, 12432, 1972, 246, 16, 2;
80, 553598, 624588, 291158, 89762, 19066, 3104, 374, 30, 6;
96, 1164192, 1305260, 615048, 189910, 41094, 6654, 844, 72, 12;
102, 1491314, 1664362, 788138, 243924, 52438, 8502, 1080, 112, 2, 2;
120, 2887184, 3203244, 1529870, 474822, 102482, 16490, 2002, 206, 22;
128, 3752194, 4153544, 1987610, 617634, 133288, 21374, 2698, 278, 42;
140, 5393824, 5962776, 2855524, 889822, 191612, 31128, 3926, 438, 26;
.
.

Scott R. Shannon, Image for n = 11.

Cf. A359116 (vertices), A359117 (regions), A359118 (edges), A358889, A006842, A006843, A005728, A331451.

Sum of row n = A359117(n).

A330914 Irregular table read by rows: Take a semicircular polygon with all vertices mutually connected by straight line segments, as in A333642. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

Original entry on oeis.org

2, 7, 1, 16, 4, 30, 10, 3, 52, 20, 8, 79, 47, 10, 3, 116, 86, 18, 4, 168, 145, 9, 2, 234, 212, 52, 12, 319, 312, 80, 17, 2, 430, 446, 96, 18, 2, 551, 616, 173, 28, 5, 730, 792, 248, 44, 6, 960, 1035, 167, 25, 1148, 1384, 422, 66, 20

Offset: 1

Scott R. Shannon and N. J. A. Sloane, May 02 2020

See A333642 for a precise definition of the polygon and images.

			Table begins:
2;
7,1;
16,4;
30,10,3;
52,20,8;
79,47,10,3;
116,86,18,4;
168,145,9,2;
234,212,52,12;
319,312,80,17,2;
430,446,96,18,2;
551,616,173,28,5;
730,792,248,44,6;
960,1035,167,25;
1148,1384,422,66,20;
1427,1745,552,108,12;
1784,2154,648,120,14;
2179,2618,927,164,27,1
2652,3200,1088,244,36;
3237,3842,1170,218,26,3,2;

Cf. A333642 (regions), A330911 (edges), A330913 (vertices), A331932, A333025, A331451.

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.

cp's OEIS Frontend

A342236 a(n) is the smallest m such that a regular m-gon with all diagonals drawn contains a cell with n sides, as in A342222, but for odd m the central m-sided polygon is not considered. Otherwise a(n) = -1 if no such m exists.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A333037 Table read by rows: T(n,k) = number of k-sided polygons in an equal-armed cross with arms of length n (see Comments in A331456 for definition) for k = 3,4,5,6,7.

Views

Author

Keywords

Comments

Examples

Crossrefs

A341730 Maximum number of sides in any cell in a regular 2n-gon with all diagonals drawn (cf. A007678).

Views

Author

Keywords

Comments

Links

Crossrefs

A333458 Irregular table read by rows: Take a diagonal-edged (or diamond-shaped) checkerboard with all diagonals drawn, as in A333434. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A342153 Irregular table read by rows: Take a vesica piscis with all diagonals drawn, as in A341877. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A359119 Irregular table read by rows: T(n,k) is the number of k-gons, k>=2, in the Farey Ring graph FR(n) defined in A359116.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A330914 Irregular table read by rows: Take a semicircular polygon with all vertices mutually connected by straight line segments, as in A333642. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

Views

Author

Keywords

Comments

Examples

Crossrefs

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs