A135565 -id:A135565 - cp's OEIS frontend

A335646 Irregular table read by rows: row n gives the number of 5-gon to k-gon contacts, with k>=5, for a regular n-gon with all diagonals drawn.

0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 39, 42, 15, 0, 15, 0, 102, 18, 190, 38, 19, 20, 80, 273, 210, 21, 154, 44, 529, 322, 69, 144, 750, 350, 598, 156, 26, 1215, 432, 81, 560, 56, 928, 406, 29, 0, 0, 0, 29, 300, 60, 2139, 248, 93, 1568, 704, 64, 1782, 792, 132

Offset: 3

Scott R. Shannon, Aug 23 2020

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 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.

			The table begins:
.
0;
0;
0;
0;
0;
0;
0;
0;
11;
0;
39;
42;
15,0,15;
0;
102;
18;
190,38,19;
20,80;
273,210,21;
154,44;
529,322,69;
144;
750,350;
598,156,26;
1215,432,81;
560,56;
928,406,29,0,0,0,29;
300,60;
2139,248,93;
1568,704,64;
1782,792,132;

Scott R. Shannon, Image for n=11 with edge-number coloring. This is the first n-gon with 5-gon to 5-gon contacts.
Scott R. Shannon, Image for n=19 with edge-number coloring. This is the first n-gon with 5-gon to 6-gon contacts. It is also the first n-gon with 7-gon to 7-gon contacts.
Scott R. Shannon, Image for n=15 with edge-number coloring. This is the first n-gon with 5-gon to 7-gon contacts.
Scott R. Shannon, Image for n=29 with edge-number coloring. This is the first n-gon with 5-gon to 11-gon contacts.
Scott R. Shannon, Image for n=29 with random distance-based coloring.

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

A337330 Irregular table read by rows: row n gives the number of 6-gon to k-gon contacts, with k>=6, for a regular n-gon with all diagonals drawn, with n>=25.

Original entry on oeis.org

50, 0, 108, 0, 0, 0, 124, 32, 66, 136, 70, 144, 148, 76, 390, 120, 328, 82, 42, 86, 86, 0, 540, 92, 92, 188, 94, 94, 0, 196, 98, 750, 100, 816, 416, 104, 1272, 432, 220, 110, 728, 570, 570, 406, 348, 116, 1062, 354, 300, 854, 366, 122, 1488, 124, 1512, 252, 126, 576, 2080, 130, 260, 2112

Offset: 25

Scott R. Shannon, Aug 23 2020

For n=3 to n=24 there are no n-gons that have 6-gon to k-gon contacts, where k>=6, so the table starts at n=25.
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 A007678 for the number of regions and images of other n-gons.

			The table begins:
.
50;
0;
108;
0;
0;
0;
124;
32;
66;
136;
70;
144;
148;
76;
390;
120;
328, 82;
42;

Scott R. Shannon, Image for n=25 with edge-number coloring. This is the first n-gon with 6-gon to 6-gon contacts.
Scott R. Shannon, Image for n=25 with random distance-based coloring.

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

a(34) and beyond from Scott R. Shannon, Jan 11 2021

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.

Original entry on oeis.org

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

A351045 Irregular table read by rows: row n gives the number of edges with k facing edges for a regular n-gon with all diagonals drawn, with n>=3 and k>=2.

Original entry on oeis.org

3, 4, 0, 4, 5, 0, 10, 0, 5, 6, 0, 18, 12, 6, 7, 0, 28, 14, 21, 14, 7, 8, 0, 56, 48, 24, 9, 0, 54, 54, 72, 72, 18, 0, 9, 10, 0, 80, 160, 120, 20, 11, 0, 88, 154, 198, 198, 55, 0, 0, 0, 11, 12, 0, 240, 336, 168, 13, 0, 130, 260, 507, 390, 91, 104, 0, 0, 0, 0, 13, 14, 0, 266, 616, 644, 140, 42

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Jan 30 2022

The number of facing edges for a given edge is the number of other edges in the one (for edges on the outside of the n-gon) or two polygons that the edge forms a part of. For example, for an edge shared between two adjoined triangles the number of facing edges is four, as it faces two edges in each of the two triangles it forms a part of.
All edges that are on the outside of the n-gon have two facing edges as any such edge belongs to only one (interior) triangle. Thus T(n,2) = n. For odd n the central created n-gon, see A342222, is surrounded by triangles, thus the edges that form this central n-gon have (n-1)+(3-1) = n+1 facing edges, thus T(n,n+1) >= n.
For all n-gons with even n, or odd n if the central n-gon is ignored, the maximum k for which row(n,k) > 0 is unknown, although it is clearly related to the maximum sided cell for all n-gons; see A349784.

			A hexagon with all diagonals drawn has six edges (those on the outside of the hexagon) which form one side of a single triangle and thus face two edges, eighteen edges that adjoin two triangles and thus face four edges, twelve edges that adjoin a triangle and a quadrilateral and thus face five edges, and six edges that adjoin two quadrilaterals and thus face six edges. Thus the row for n = 6 is [6, 0, 18, 12, 6]. See the attached image.
The table begins:
3;
4, 0, 4;
5, 0, 10, 0, 5;
6, 0, 18, 12, 6;
7, 0, 28, 14, 21, 14, 7;
8, 0, 56, 48, 24;
9, 0, 54, 54, 72, 72, 18, 0, 9;
10, 0, 80, 160, 120, 20;
11, 0, 88, 154, 198, 198, 55, 0, 0, 0, 11;
12, 0, 240, 336, 168;
13, 0, 130, 260, 507, 390, 91, 104, 0, 0, 0, 0, 13;
14, 0, 266, 616, 644, 140, 42;
15, 0, 180, 600, 945, 630, 435, 0, 15, 0, 0, 0, 0, 0, 15;
16, 0, 448, 1056, 960, 576, 32;
17, 0, 238, 816, 1853, 1224, 425, 272, 34, 0, 0, 0, 0, 0, 0, 0, 17;
18, 0, 900, 1836, 1314, 108, 144;
19, 0, 304, 1520, 2717, 2128, 798, 304, 95, 0, 19, 0, 0, 0, 0, 0, 0, 0, 19;
20, 0, 1000, 2120, 3280, 1600, 100, 240;
21, 0, 378, 2352, 4494, 3276, 1365, 252, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21;
22, 0, 1056, 3828, 5258, 1716, 374, 396, 132;
.
.
See the linked file for the table n = 3..100.

Scott R. Shannon, Table for n = 3..100.
Scott R. Shannon, Image of the edges for n = 6.
Scott R. Shannon, Image of the edges for n = 9.
Scott R. Shannon, Image of the edges for n = 15.
Scott R. Shannon, Image of the edges for n = 25.
Scott R. Shannon, Image of the edges for n = 30.

Cf. A135565, A342222, A349784, A344899, A342236, A342222, A342236.

Sum of row n = A135565(n).
T(n,2) = n.
T(n,n+1) >= n for odd n.

A349968 Number of edges in regular n-gon after 2 generations of mitosis.

Original entry on oeis.org

3, 8, 35, 66, 308, 232, 1305, 900, 3399, 1428, 9061, 4704, 15345, 9424, 30294, 11376, 48773, 30840, 66738, 49148, 108330, 46320, 153825, 105690, 212355, 148876, 337328, 108330, 399404, 264320, 514866, 365874, 657265, 393264, 844969, 598272, 1017510, 774520, 1319257, 670152, 1641224, 1171192

Offset: 3

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

nonn

See A350000 for further details.

Cf. A349807 (cells), A349967 (vertices), A350000, A135565.

A371255 Number of (curved) edges 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, 2, 9, 8, 40, 24, 168, 152, 477, 490, 1199, 912, 2418, 2464, 4230, 4464, 7769, 6894, 12369, 12400, 18606, 19008, 27784, 26376, 39575, 39390, 54027, 54432, 73254, 68340, 96410, 96800, 124443, 125222, 159005, 156168, 200540, 201932, 248508, 250120

Offset: 1

Scott R. Shannon, Mar 16 2024

nonn

See A371253 and A371254 for images of the circles.

Cf. A371253 (regions), A371254 (vertices), A371274 (k-gons), A135565, A358783, A359047, A359254.

a(n) = A371253(n) + A371254(n) - 1 by Euler's formula.

A330911 The number of edges formed by straight line segments mutually connecting all vertices of a semicircular polygon defined in A333642.

Original entry on oeis.org

5, 15, 35, 76, 142, 251, 408, 576, 947, 1367, 1845, 2600, 3460, 4011, 5822, 7386, 9023, 11423, 13967, 16242, 20330, 24235, 28222, 33686, 39327, 44967, 52733, 60608, 67383, 78947, 89530, 100040, 113885, 127791, 141925, 159356, 177158, 194895, 217232, 239662

Offset: 1

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

nonn

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

Lars Blomberg, Table of n, a(n) for n = 1..100

Cf. A333642 (regions), A330913 (vertices), A330914 (n-gons), A333278, A333027, A135565.

a(21) and beyond from Lars Blomberg, May 03 2020

A342152 The number of edges on a vesica piscis formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

Original entry on oeis.org

2, 8, 42, 148, 438, 936, 2010, 3462, 6038, 8816, 14606, 20504, 29854, 39790, 54618, 70142, 92662, 115718, 147494, 177500, 223506, 267872, 326142, 384274, 460302, 535896, 631886, 726674, 848126, 965592, 1115194, 1259926, 1440558, 1616940, 1833130, 2042602, 2300498, 2549756, 2851626, 3139854

Offset: 1

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

nonn

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.

Wikipedia, Vesica piscis.

Cf. A341877 (regions), A341878 (vertices), A342153 (n-gons), A135565, A332376, A340613, A340687.

a(n) = A341877(n) + A341878(n) - 1.

A353876 Irregular table read by rows: for each internal vertex of a regular n-gon with all diagonals drawn remove all the edges connected directly to the vertex and then count the number of sides in the polygon that surrounds it; row n gives the number of resulting k-sided polygons, for k>=4, for all internal vertices.

Original entry on oeis.org

0, 1, 0, 0, 5, 0, 6, 6, 0, 0, 0, 0, 0, 1, 0, 7, 0, 14, 0, 7, 7, 0, 8, 24, 8, 9, 0, 9, 18, 18, 0, 63, 0, 18, 0, 10, 70, 30, 20, 10, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 11, 44, 33, 55, 143, 11, 22, 0, 11, 12, 24, 144, 24, 60, 0, 36, 0, 1, 0, 13, 78, 39, 130, 260, 91, 65, 26, 0, 0, 13

Offset: 3

Scott R. Shannon, May 09 2022

Numerous patterns are found in the values of the k-gons for different n. For example for n = 4*m + 2, with m>=1, there is one maximum sided k-gon with 2*n edges. For n = 4*m, with m>=3, there is one maximum sided k-gon with n edges. For odd n, where n>=11, there is n maximum sided k-gons with n+2 edges.
The 8-gon appears to be unique in that there is 9 maximum sided k-gons, k=8, which is not 1 or a multiple of 8.
Only a limit number of even-n n-gons have vertex-surrounding polygons with 4 edges, the minimum possible value. See A353991.

			The 7-gon has seven internal vertices surrounded by polygons with 5 edges, fourteen internal vertices surrounded by polygons with 7 edges, seven internal vertices surrounded by polygons with 9 edges, and seven internal vertices surrounded by polygons with 10 edges, so row 7 is [0, 7, 0, 14, 0, 7, 7].
The table begins:
0;
1;
0, 0, 5;
0, 6, 6, 0, 0, 0, 0, 0, 1;
0, 7, 0, 14, 0, 7, 7;
0, 8, 24, 8, 9;
0, 9, 18, 18, 0, 63, 0, 18;
0, 10, 70, 30, 20, 10, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
0, 11, 44, 33, 55, 143, 11, 22, 0, 11;
12, 24, 144, 24, 60, 0, 36, 0, 1;
0, 13, 78, 39, 130, 260, 91, 65, 26, 0, 0, 13;
0, 14, 182, 196, 168, 126, 56, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \
                                                                       0, 0, 1;
0, 15, 120, 90, 345, 525, 135, 105, 15, 0, 0, 0, 0, 15;
0, 32, 256, 240, 480, 224, 96, 16, 32, 0, 0, 0, 1;
.

Scott R. Shannon, Image of the 7-gon. In this and other images the vertex color is based on the surrounding polygon edge count shown in the key.
Scott R. Shannon, Image of the 8-gon.
Scott R. Shannon, Image of the 9-gon.
Scott R. Shannon, Image of the 10-gon.
Scott R. Shannon, Image of the 12-gon.
Scott R. Shannon, Table for n=3..100.

Cf. A353991, A007569, A007678, A135565, A351045.

Sum of terms in row n = A007569(n) - n.

A357254 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of edges in an n-gon when k internal n-gons are drawn between the n*k points that divide each side into k+1 equal parts.

Original entry on oeis.org

3, 9, 4, 27, 12, 5, 57, 36, 15, 6, 99, 76, 45, 18, 7, 135, 132, 95, 54, 21, 8, 219, 180, 165, 114, 63, 24, 9, 297, 292, 255, 198, 133, 72, 27, 10, 351, 348, 365, 306, 231, 152, 81, 30, 11, 489, 516, 495, 438, 357, 264, 171, 90, 33, 12, 603, 604, 645, 594, 511, 408, 297, 190, 99, 36, 13

Offset: 3

Scott R. Shannon, Sep 20 2022

Conjecture: the only n-gons that contain non-simple intersections are the 3-gon (triangle), 4-gon (square), and 6-gon (hexagon).

			The table begins:
   3,  9,  27,  57,  99, 135,  219,  297,  351,  489,  603,  645,  867, 1017, ...
   4, 12,  36,  76, 132, 180,  292,  348,  516,  604,  804,  892, 1156, 1284, ...
   5, 15,  45,  95, 165, 255,  365,  495,  645,  815, 1005, 1215, 1445, 1695, ...
   6, 18,  54, 114, 198, 306,  438,  594,  774,  942, 1206, 1422, 1734, 2034, ...
   7, 21,  63, 133, 231, 357,  511,  693,  903, 1141, 1407, 1701, 2023, 2373, ...
   8, 24,  72, 152, 264, 408,  584,  792, 1032, 1304, 1608, 1944, 2312, 2712, ...
   9, 27,  81, 171, 297, 459,  657,  891, 1161, 1467, 1809, 2187, 2601, 3051, ...
  10, 30,  90, 190, 330, 510,  730,  990, 1290, 1630, 2010, 2430, 2890, 3390, ...
  11, 33,  99, 209, 363, 561,  803, 1089, 1419, 1793, 2211, 2673, 3179, 3729, ...
  12, 36, 108, 228, 396, 612,  876, 1188, 1548, 1956, 2412, 2916, 3468, 4068, ...
  13, 39, 117, 247, 429, 663,  949, 1287, 1677, 2119, 2613, 3159, 3757, 4407, ...
  14, 42, 126, 266, 462, 714, 1022, 1386, 1806, 2282, 2814, 3402, 4046, 4746, ...
  15, 45, 135, 285, 495, 765, 1095, 1485, 1935, 2445, 3015, 3645, 4335, 5085, ...
  ...
See the attached text file for further examples.
See A356984, A357058, A357196 for images of the n-gons.

Scott R. Shannon, Extended table for n = 3..50, k = 0..75.

Cf. A357216 (regions), A357235 (vertices), A357008 (triangle), A357061 (square), A357198 (hexagon), A356984, A357058, A357196, A135565, A344899.

T(n,k) = A357216(n,k) + A357235(n,k) - 1 by Euler's formula.
T(n,0) = n.
T(n,1) = 3n.
Conjectured formula for all columns for n >= 7: T(n,k) = 2n*k^2 + n.
T(3,k) = A357008(k).
T(4,k) = A357061(k).
T(6,k) = A357198(k).
Conjectured formula for all rows except for n = 3, 4, 6: T(n,k) = 2n*k^2 + n.

cp's OEIS Frontend

A335646 Irregular table read by rows: row n gives the number of 5-gon to k-gon contacts, with k>=5, for a regular n-gon with all diagonals drawn.

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A337330 Irregular table read by rows: row n gives the number of 6-gon to k-gon contacts, with k>=6, for a regular n-gon with all diagonals drawn, with n>=25.

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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.

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A351045 Irregular table read by rows: row n gives the number of edges with k facing edges for a regular n-gon with all diagonals drawn, with n>=3 and k>=2.

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A349968 Number of edges in regular n-gon after 2 generations of mitosis.

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A371255 Number of (curved) edges 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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A330911 The number of edges formed by straight line segments mutually connecting all vertices of a semicircular polygon defined in A333642.

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A342152 The number of edges on a vesica piscis formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

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A357254 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of edges in an n-gon when k internal n-gons are drawn between the n*k points that divide each side into k+1 equal parts.

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