A062361 -id:A062361 - cp's OEIS frontend

A333654 Irregular table read by rows: row n gives the number of 3-gon to k-gon contacts, with k>=3, for a regular n-gon with all diagonals drawn.

0, 4, 10, 0, 5, 18, 12, 28, 14, 21, 0, 7, 56, 48, 54, 54, 54, 36, 0, 0, 9, 180, 160, 30, 88, 154, 88, 88, 0, 0, 0, 0, 11, 240, 336, 130, 260, 299, 182, 0, 104, 0, 0, 0, 0, 13, 266, 616, 266, 180, 600, 375, 300, 90, 0, 0, 0, 0, 0, 0, 0, 15, 448, 1056, 320, 256

Offset: 3

Scott R. Shannon, Aug 23 2020

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.

			The table begins:
.
0;
4;
10,0,5;
18,12;
28,14,21,0,7;
56,48;
54,54,54,36,0,0,9;
180,160,30;
88,154,88,88,0,0,0,0,11;
240,336;
130,260,299,182,0,104,0,0,0,0,13;
266,616,266;
180,600,375,300,90,0,0,0,0,0,0,0,15;
448,1056,320,256;
238,816,935,578,85,238,0,0,0,0,0,0,0,0,17;
900,1836,324,108,126;
304,1520,1463,798,304,228,0,0,0,0,0,0,0,0,0,0,19;
1000,2120,1740,560,0,160;
378,2352,2016,1554,84,0,0,0,0,0,0,0,0,0,0,0,0,0,21;
1056,3828,2310,308,176,220;
460,2852,4117,2530,644,138,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23;
2352,5856,2376,288;
550,4550,4800,3350,900,250,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25;
1950,6708,5772,1612,650,208;
648,6372,6399,4968,432,810,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27;
2380,10024,5712,3584,616,224;
754,8642,8497,6148,783,928,0,870,232,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29;
6330,14640,5340,2340;
868,11532,11191,7378,806,1302,0,496,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31;
3712,16384,12000,6016,1248;
990,14454,14619,12540,2079,1782,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,33;

Scott R. Shannon, Image for n=4 with edge-count coloring. This is the first n-gon with 3-gon to 3-gon contacts.
Scott R. Shannon, Image for n=6 with edge-count coloring. This is the first n-gon with 3-gon to 4-gon contacts.
Scott R. Shannon, Image for n=7 with edge-count coloring. This is the first n-gon with 3-gon to 5-gon contacts, other than n=5 which has an internal 5-gon surrounded by 3-gons by its construction.
Scott R. Shannon, Image for n=9 with edge-count coloring. This is the first n-gon with 3-gon to 6-gon contacts.
Scott R. Shannon, Image for n=15 with edge-count coloring. This is the first n-gon with 3-gon to 7-gon contacts, other than n=7 which has an internal 7-gon surrounded by 3-gons by its construction.
Scott R. Shannon, Image for n=15 with random distance-based coloring.
Scott R. Shannon, Image for n=13 with edge-count coloring. This is the first n-gon with 3-gon to 8-gon contacts.
Scott R. Shannon, Image for n=29 with edge-count coloring. This is the first n-gon with 3-gon to 10-gon and 3-gon to 11-gon contacts.

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

For odd n, with n > 3, row(n,n) = n as the n-gon contains a smaller n-gon at its center surrounded by 3-gons.

A335614 Irregular table read by rows: row n gives the number of 4-gon to k-gon contacts, with k>=4, for a regular n-gon with all diagonals drawn.

Original entry on oeis.org

0, 0, 0, 6, 0, 5, 24, 18, 36, 18, 90, 20, 110, 110, 44, 168, 208, 208, 52, 378, 140, 570, 330, 330, 640, 320, 32, 918, 646, 238, 34, 34, 990, 1254, 1330, 304, 38, 76, 1540, 1040, 80, 2478, 1722, 1008, 42, 2948, 1408, 44, 132, 132, 2852, 2254, 1012, 92, 46, 3624, 1536, 144

Offset: 3

Scott R. Shannon, Aug 23 2020

See A333654 for the number of 3-gon to k-gon contacts, with k>=3.
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.

			The table begins:
.
0;
0;
0;
6;
0,5;
24;
18,36,18;
90,20;
110,110,44;
168;
208,208,52;
378,140;
570,330,330;
640,320,32;
918,646,238,34,34;
990;
1254,1330,304,38,76;
1540,1040,80;
2478,1722,1008,42;
2948,1408,44,132,132;
2852,2254,1012,92,46;
3624,1536,144;
4650,3600,1450,150,150;
4784,3380,572,52;
6696,4968,2160,54,54;
7280,3752,2072,168;
8990,7308,3016,0,232,0,0,58;
7770,1200,300;
13020,9920,3472,186,434,0,124;
12160,8576,2048,256;
14784,11418,5742,792,330;

Scott R. Shannon, Image for n=6 with edge-number coloring. This is the first n-gon with 4-gon to 4-gon contacts.
Scott R. Shannon, Image for n=7 with edge-number coloring. This is the first n-gon with 4-gon to 5-gon contacts.
Scott R. Shannon, Image for n=9 with edge-number coloring. This is the first n-gon with 4-gon to 6-gon contacts.
Scott R. Shannon, Image for n=17 with edge-number coloring. This is the first n-gon with 4-gon to 7-gon and 4-gon to 8-gon contacts.
Scott R. Shannon, Image for n=31 with edge-number coloring. This is the first n-gon with 4-gon to 10-gon contacts.
Scott R. Shannon, Image for n=31 with random distance-based coloring.
Scott R. Shannon, Image for n=29 with edge-number coloring. This is the first n-gon with 4-gon to 11-gon contacts.
Scott R. Shannon, Image for n=18 with edge-number coloring. This is the only n-gon from n=13 to n=33 that has only 4-gon to 3-gon and 4-gon to 4-gon contacts. It is likely the last n-gon with this property.
Scott R. Shannon, Image for n=18 with random distance-based coloring.

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

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.

Original entry on oeis.org

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

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.

A300552 Number of 4-cycles in the n-polygon diagonal intersection graph.

Original entry on oeis.org

0, 5, 10, 24, 35, 80, 90, 170, 220, 420, 364, 644, 780, 1120, 1139, 1854, 1748, 2580, 2898, 3916, 3450, 6072, 5350, 7046, 7398, 10024, 10063, 14250, 13795, 16608, 16566, 21114, 21595, 26316, 26640, 32718, 32370, 39520, 38622, 52416, 47558, 56232, 57240

Offset: 3

Eric W. Weisstein, Mar 08 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 3..100
Eric Weisstein's World of Mathematics, Graph Cycle
Eric Weisstein's World of Mathematics, Polygon Diagonal Intersection Graph

Cf. A062361 (3-cycles), A300553 (5-cycles), A300554 (6-cycles).

Terms a(21) and beyond from Andrew Howroyd, Mar 11 2018

A300553 Number of 5-cycles in the n-polygon diagonal intersection graph.

Original entry on oeis.org

0, 4, 12, 36, 56, 120, 135, 250, 297, 684, 520, 910, 945, 1616, 1428, 2826, 2489, 3500, 3654, 5346, 4899, 9000, 7175, 10010, 9909, 14028, 13050, 21960, 17577, 23552, 21582, 28798, 28385, 38952, 36778, 47044, 43992, 57680, 51455, 77238, 66865, 81576

Offset: 3

Eric W. Weisstein, Mar 08 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 3..100
Eric Weisstein's World of Mathematics, Graph Cycle
Eric Weisstein's World of Mathematics, Polygon Diagonal Intersection Graph

A062361 (3-cycles), A300552 (4-cycles), A300554 (6-cycles).

Terms a(21) and beyond from Andrew Howroyd, Mar 11 2018

A300554 Number of 6-cycles in the n-polygon diagonal intersection graph.

Original entry on oeis.org

0, 0, 35, 91, 98, 272, 279, 530, 561, 1440, 1079, 1904, 1890, 3360, 3128, 6066, 4579, 7580, 6993, 10648, 10212, 18696, 13875, 19968, 18981, 26936, 25056, 47130, 32953, 44864, 41184, 55386, 52500, 74808, 63714, 86564, 84045, 105840, 102049, 149982, 119196

Offset: 3

Eric W. Weisstein, Mar 08 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 3..100
Eric Weisstein's World of Mathematics, Polygon Diagonal Intersection Graph
Eric Weisstein's World of Mathematics, Graph Cycle

Cf. A062361 (3-cycles), A300552 (4-cycles), A300553 (5-cycles).

Terms a(21) and beyond from Andrew Howroyd, Mar 11 2018

A187782 Number of different kinds of polygons in a regular n-gon with all diagonals drawn.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 5, 3, 5, 2, 6, 3, 6, 4, 7, 5, 7, 5, 6, 6, 7, 4, 7, 6, 7, 6, 9, 4, 8, 5, 7, 6, 8, 6, 8, 6, 7, 7, 9, 6, 8, 8, 8, 6, 8, 7, 8, 7, 10, 6, 9, 7, 9, 7, 9, 7, 10, 7

Offset: 3

Martin Renner, Jan 05 2013

			a(5) = 2 since the 11 regions of the regular pentagon built by all diagonals consist of two different kinds of polygons, i.e., 10 triangles and 1 pentagon.
a(6) = 2 since the 24 regions of the regular hexagon built by all diagonals consist of two different kinds of polygons, i.e., 18 triangles and 6 quadrilaterals.
a(7) = 4 since the 50 regions of the regular heptagon built by all diagonals consist of four different kinds of polygons, i.e., 35 triangles, 7 quadrilaterals, 7 pentagons and 1 heptagon.

Sascha Kurz, Anzahl von Dreiecken eines regelmäßigen n-Ecks.
Bjorn Poonen and Michael Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics 11 (1998), nr. 1, pp. 135-156; doi: 10.1137/S0895480195281246, arXiv: math.MG/9508209.
Eric Weisstein's World of Mathematics, Regular Polygon Division by Diagonals.

Cf. A007678, A062361, A067151, A067152, A067153, A067154, A067155, A067156, A067157, A067158, A067159.

a(45)-a(60) from Christopher Scussel, Jun 24 2023

A217749 Number of triangular regions in the exterior of a regular n-gon with all diagonals drawn.

Original entry on oeis.org

0, 0, 0, 0, 7, 24, 45, 70, 121, 180, 273, 406, 555, 784, 1071

Offset: 3

Martin Renner, Mar 23 2013

nonn

Cf. A062361, A217746.

cp's OEIS Frontend

A333654 Irregular table read by rows: row n gives the number of 3-gon to k-gon contacts, with k>=3, for a regular n-gon with all diagonals drawn.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A335614 Irregular table read by rows: row n gives the number of 4-gon to k-gon contacts, with k>=4, for a regular n-gon with all diagonals drawn.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

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

A300552 Number of 4-cycles in the n-polygon diagonal intersection graph.

Views

Author

Keywords

Links

Crossrefs

Extensions

A300553 Number of 5-cycles in the n-polygon diagonal intersection graph.

Views

Author

Keywords

Links

Crossrefs

Extensions

A300554 Number of 6-cycles in the n-polygon diagonal intersection graph.

Views

Author

Keywords

Links

Crossrefs

Extensions

A187782 Number of different kinds of polygons in a regular n-gon with all diagonals drawn.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A217749 Number of triangular regions in the exterior of a regular n-gon with all diagonals drawn.

Views

Author

Keywords

Crossrefs