A331450 -id:A331450 - cp's OEIS frontend

A342222 a(n) is the smallest m such that a regular m-gon with all diagonals drawn contains a cell with n sides, or a(n) = -1 if no such m exists.

3, 6, 5, 9, 7, 13, 9, 29, 11, 40, 13, 43, 15, 212, 17, 231, 19

Offset: 3

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

Theorem: If n is odd then a(n) = n.
Proof. (i) If n is odd then the central cell in a regular n-gon with all diagonals drawn is a smaller regular n-gon. So if n is odd, then a(n) <= n.
(ii) Suppose a convex m-gon, not necessarily regular, with all diagonals drawn has a cell with e edges. Each edge when extended meets two vertices, so at most 2e vertices are involved in defining the boundary of that cell.
On the other hand no vertex can define more than two edges of the cell, so 2e <= 2m, so e <= m. So to get an n-sided cell, we need at least n vertices. So a(n) >= n. QED.
If a(20) > 0 it is greater than 765 - Scott R. Shannon, Nov 30 2021

			Examining the images in A007678, for example Michael Rubinstein's illustration, or the images shown here, we see that the first occurrence of a five-sided cell is for m = 5, so a(5) = 5. The first time we see a four-sided cell is for m = 6, so a(4) = 6.

Martin Balko, Anna Brötzner, Fabian Klute, and Josef Tkadlec, Faces in Rectilinear Drawings of Complete Graphs, 40th European Workshop on Computational Geometry, Ioannina, Greece, March 13-15, 2024. See pp. 4, 7.
Scott R. Shannon, Image for a(3) = 3.
Scott R. Shannon, Image for a(4) = 6.
Scott R. Shannon, Image for a(5) = 5.
Scott R. Shannon, Image for a(6) = a(9) = 9.
Scott R. Shannon, Image for a(7) = 7.
Scott R. Shannon, Image for a(8) = a(13) = 13.
Scott R. Shannon, Image for a(10) = 29.
Scott R. Shannon, Image for a(11) = 11.
Scott R. Shannon, Image for a(12) = 40.
Scott R. Shannon, Image for a(14) = 43.
Scott R. Shannon, Image for a(15) = 15.

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

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

a(16)-a(19) added by Scott R. Shannon, Mar 14 2021

A344938 Irregular triangle read by rows: T(n,k) = number of k-sided polygons formed when every pair of vertices of a regular n-gon are joined by an infinite line, for k = 3, 4, ..., max_k.

Original entry on oeis.org

1, 4, 15, 0, 1, 36, 6, 70, 21, 7, 0, 1, 112, 64, 189, 108, 36, 18, 0, 0, 1, 270, 220, 50, 407, 352, 110, 55, 0, 0, 0, 0, 1, 624, 528, 884, 689, 325, 91, 0, 26, 0, 0, 0, 0, 1, 1162, 1092, 266, 14, 1530, 1545, 480, 270, 45, 0, 0, 0, 0, 0, 0, 0, 1, 2080, 2032, 416, 80

Offset: 3

Scott R. Shannon, Jun 03 2021

See A344857 for examples and images of the polygons.

			A pentagon with all vertices connected forms 10 triangles inside the pentagon, 5 triangles outside the pentagon, giving 15 triangles in all, and 1 smaller pentagon inside the pentagon, so row 3 is [15,0,1].
The table begins:
1;
4;
15,0,1;
36,6;
70,21,7,0,1;
112,64;
189,108,36,18,0,0,1;
270,220,50;
407,352,110,55,0,0,0,0,1;
624,528;
884,689,325,91,0,26,0,0,0,0,1;
1162,1092,266,14;
1530,1545,480,270,45,0,0,0,0,0,0,0,1;
2080,2032,416,80;
2567,2754,1003,374,17,68,0,0,0,0,0,0,0,0,1;
3402,3366,180,18,18;
3952,4807,1672,475,95,76,0,19,0,0,0,0,0,0,0,0,1;
5380,5360,1580,240,0,20;
5943,7392,2583,1260,21,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
7590,9020,2310,132,132,66;
9430,9775,4508,1518,253,46,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
11304,12288,2280,144;
13025,14650,6250,2375,200,75,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
16042,16952,5954,728,260,52;
17064,22464,7884,2700,567,189,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
21616,24192,7056,2016,168,28;
23751,29319,11281,3828,348,319,0,87,29,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
29880,29010,4140,540;
30814,39370,15314,5177,341,496,0,62,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
37440,42624,14240,3008,544,64;
41481,49335,19305,7854,891,363,66,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;

Cf. A344857 (total number of polygons), A344899 (number of edges), A146212 (number of vertices), A344866, A344311, A007678, A331450 (number of k-gons inside the regular n-gon).

Sum of row(n) = A344857(n) = A344311(n) + A007678(n).

A341729 Maximum number of sides in any cell in a regular n-gon with all diagonals drawn (cf. A007678).

Original entry on oeis.org

3, 3, 5, 4, 7, 4, 9, 5, 11, 4, 13, 5, 15, 6, 17, 7, 19, 8, 21, 8, 23, 6, 25, 8, 27, 8, 29, 6, 31, 7, 33, 8, 35, 8, 37, 8, 39, 12, 41, 8, 43, 10, 45, 8, 47, 10, 49, 10, 51, 8, 53, 10, 55, 10, 57, 10, 59, 9, 61, 10, 63, 10, 65, 9, 67, 10, 69, 10, 71, 10, 73, 10, 75, 12, 77, 10, 79, 10

Offset: 3

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

nonn

For a(2*n) see A341730.
Theorem: a(2*n+1) = 2*n+1. For the proof see A342222.
It would be nice to have a bigger b-file.

Scott R. Shannon, Table of n, a(n) for n = 3..765

Cf. A007678, A062361, A288177, A331450, A331451, A342222.

Bisections: A144396, A341730.

a(141) and beyond from Scott R. Shannon, Nov 30 2021

A371376 Irregular table read by rows: place n equally spaced points around the circumference of a circle and then, for each pair of points, draw two distinct circles, whose radii are the same as the first circle, such that both points lie on their circumferences. T(n,k), k>=2, gives the number of k-sided regions formed.

Original entry on oeis.org

1, 6, 3, 8, 0, 1, 15, 30, 5, 1, 18, 30, 14, 147, 35, 7, 7, 1, 8, 152, 48, 8, 0, 0, 1, 27, 351, 171, 36, 27, 10, 390, 200, 10, 40, 0, 0, 0, 1, 22, 693, 649, 33, 77, 0, 0, 0, 0, 1, 12, 780, 408, 0, 48, 26, 1404, 1183, 234, 169, 0, 0, 0, 0, 0, 0, 1, 14, 1498, 1274, 224, 154, 14, 14, 0, 0, 0, 0, 0, 1

Offset: 2

Scott R. Shannon, Mar 20 2024

See A371373 and A371374 for images of the graphs.

			The table begins:
1;
6, 3;
8, 0, 1;
15, 30, 5, 1;
18, 30;
14, 147, 35, 7, 7, 1;
8, 152, 48, 8, 0, 0, 1;
27, 351, 171, 36, 27;
10, 390, 200, 10, 40, 0, 0, 0, 1;
22, 693, 649, 33, 77, 0, 0, 0, 0, 1;
12, 780, 408, 0, 48;
26, 1404, 1183, 234, 169, 0, 0, 0, 0, 0, 0, 1;
14, 1498, 1274, 224, 154, 14, 14, 0, 0, 0, 0, 0, 1;
45, 2310, 2400, 390, 255, 15;
16, 2736, 2032, 656, 320, 0, 32, 0, 0, 0, 0, 0, 0, 0, 1;
34, 3978, 4097, 969, 493, 17, 34, 0, 0, 0, 0, 0, 0, 0, 0, 1;
18, 4410, 3078, 972, 468, 36, 18;
76, 6365, 6365, 1596, 855, 95, 76, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
20, 6840, 6000, 2100, 780, 60, 100, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
63, 8946, 10395, 2751, 924, 126, 147;
22, 10076, 9218, 3674, 1166, 22, 132, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
46, 13156, 14996, 4347, 1702, 92, 138, 23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
                                                                     \\ 0, 0, 1;
24, 14232, 13296, 4512, 1440, 96, 240;
100, 19075, 19850, 6975, 2675, 150, 175, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
                                                                  \\ 0, 0, 0, 1;
.
.

Cf. A371373 (vertices), A371374 (regions), A371375 (edges), A371377 (vertex crossings), A371274, A331450, A359009, A359061.

Sum of row(n) = A371374(n).

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.

Original entry on oeis.org

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

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

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.

cp's OEIS Frontend

A342222 a(n) is the smallest m such that a regular m-gon with all diagonals drawn contains a cell with n sides, or a(n) = -1 if no such m exists.

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A344938 Irregular triangle read by rows: T(n,k) = number of k-sided polygons formed when every pair of vertices of a regular n-gon are joined by an infinite line, for k = 3, 4, ..., max_k.

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A341729 Maximum number of sides in any cell in a regular n-gon with all diagonals drawn (cf. A007678).

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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.

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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.

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

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