A007569 -id:A007569 - 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.

A335102 Irregular triangle read by rows: consider the regular n-gon defined in A007678. T(n,k) (n >= 1, k >= 4+2*t where t>=0) is the number of non-boundary vertices in the n-gon at which k polygons meet.

Original entry on oeis.org

0, 0, 0, 1, 5, 12, 1, 35, 40, 8, 1, 126, 140, 20, 0, 1, 330, 228, 60, 12, 0, 1, 715, 644, 112, 0, 0, 0, 1, 1365, 1168, 208, 0, 0, 0, 0, 1, 2380, 1512, 216, 54, 54, 0, 0, 0, 1, 3876, 3360, 480, 0, 0, 0, 0, 0, 0, 1, 5985, 5280, 660, 0, 0, 0, 0, 0, 0, 0, 1, 8855, 6144, 864, 264, 24, 0, 0, 0, 0, 0, 0, 12, 12650

Offset: 1

Scott R. Shannon and N. J. A. Sloane, May 23 2020

			Table begins:
      0;
      0;
      0;
      1;
      5;
     12,    1;
     35;
     40,    8,   1;
    126;
    140,   20,   0,   1;
    330;
    228,   60,  12,   0,   1;
    715;
    644,  112,   0,   0,   0,  1;
   1365;
   1168,  208,   0,   0,   0,  0, 1;
   2380;
   1512,  216,  54,  54,   0,  0, 0, 1;
   3876;
   3360,  480,   0,   0,   0,  0, 0, 0, 1;
   5985;
   5280,  660,   0,   0,   0,  0, 0, 0, 0, 1;
   8855;
   6144,  864, 264,  24,   0,  0, 0, 0, 0, 0, 1;
  12650;
  11284, 1196,   0,   0,   0,  0, 0, 0, 0, 0, 0, 1;
  17550;
  15680, 1568,   0,   0,   0,  0, 0, 0, 0, 0, 0, 0, 1;
  23751;
  13800, 2250, 420, 180, 120, 30, 0, 0, 0, 0, 0, 0, 0, 1;
  31465;
  28448, 2464,   0,   0,   0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  40920;
  37264, 2992,   0,   0,   0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  52360;

B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG]; some typos in the published version are corrected in the revisions from 2006.
Scott R. Shannon, Image of the vertices for n=5.
Scott R. Shannon, Image of the vertices for n=6.
Scott R. Shannon, Image of the vertices for n=8.
Scott R. Shannon, Image of the vertices for n=12.
Scott R. Shannon, Image of the vertices for n=13.
Scott R. Shannon, Image of the vertices for n=16.
Scott R. Shannon, Image of the vertices for n=18.
Scott R. Shannon, Image of the vertices for n=20.
Scott R. Shannon, Image of the vertices for n=24.
Scott R. Shannon, Image of the vertices for n=30.
Index entries for sequences related to stained glass windows

Columns give A292104, A101363 (2n-gon), A101364, A101365.
Row sums give A006561.
Cf. A007569, A007678, A053126, A292105, A333275.

If n = 2t+1 is odd then the n-th row has a single term, T(2t+1, 2t+4) = binomial(2t+1,4) (these values are given in A053126).

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

A368756 Number of vertices in the hyperoctahedral (or cocktail party) graph of order n.

Original entry on oeis.org

2, 5, 17, 49, 151, 273, 693, 1249, 1711, 3525, 5529, 6777, 11711, 16133, 15937, 29121, 38227, 44561, 61985, 77041, 81423, 116165, 140997, 157649, 201211, 237125, 263449, 324689, 377359, 392185, 499789, 570241, 621255, 735493, 831537, 909097, 1048887, 1171013, 1265501, 1450081, 1608523

Offset: 1

Scott R. Shannon, Jan 04 2024

nonn

Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 6.
Scott R. Shannon, Image for n = 9.
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 15. Note this 30-gon still contains vertices with 7 chords crossing, so this maximum possible value is the same as the regular n-gon with all diagonals drawn; see A007569.
Eric Weisstein's World of Mathematics, Cocktail Party Graph.

Cf. A368755 (regions), A368757 (edges), A368758 (k-gons), A007569, A129348, A193130, A282010.

a(n) = A368757(n) - A368755(n) + 1 by Euler's formula.

A146213 Number of intersection points outside the n-gon of all lines through pairs of vertices of a regular n-gon.

Original entry on oeis.org

0, 0, 5, 18, 49, 88, 198, 300, 550, 588, 1235, 1414, 2415, 2576, 4284, 3708, 7068, 7280, 11025, 11330, 16445, 14592, 23650, 23972, 32994, 33208, 44863, 34830, 59675, 59808, 77880, 78064, 99960, 92844, 126429, 126426, 157833, 157560, 194750

Offset: 3

T. D. Noe, Oct 28 2008

nonn

These are a subset of the points counted in A146212.
Note that n divides a(n). For odd n, it appears that a(n)=n(2n^3-15n^2+34n-21)/24. [Corrected by N. J. A. Sloane, Jun 11 2025. See next line.]
Theorem: For odd n >= 3, a(n) = n*(n-1)*(n-3)*(2*n-7)/24. Proof. We now know that for n odd >= 3, A146212(n) = n*(n^3-7*n^2+15*n-1)/8. Subtracting n + binomial(n,4) (see A007569), we get the stated formula. - N. J. A. Sloane, Jun 11 2025

Jon E. Schoenfield, Table of n, a(n) for n = 3..100

Cf. A007569, A146212.

a(n) = A146212(n) - A007569(n).

More terms from Jon E. Schoenfield, Nov 11 2008

Definition clarified by N. J. A. Sloane, Jun 11 2025

A332418 The number of vertices on a decagon 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

171, 3581, 23651, 80191, 213041, 444251, 862711, 1481141, 2413721, 3701951, 5493891, 7765621, 10833601, 14589491, 19315751, 25064491, 32107771, 40337021, 50328771, 61790891, 75318371

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 09 2020

See the links in A333139 for images of the decagons.

Cf. A333139 (regions), A332417 (n-gons), A332419 (edges), A330846, A092866, A332599, A007569.

a(6)-a(21) from Lars Blomberg, May 18 2020

A332428 The number of vertices on a nonagon 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

135, 2395, 16434, 53155, 141147, 293374, 565767, 966493, 1580940, 2411533, 3581613, 5070655, 7057026, 9493435, 12594564, 16307974, 20902338, 26269597, 32760774, 40217905, 49049919, 59090671, 70803180

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 09 2020

See the links in A332421 for images of the nonagons.

Cf. A332421 (regions), A332427 (n-gons), A332429 (edges), A330846, A092866, A332599, A007569.

a(6)-a(23) from Lars Blomberg, May 16 2020

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.

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A335102 Irregular triangle read by rows: consider the regular n-gon defined in A007678. T(n,k) (n >= 1, k >= 4+2*t where t>=0) is the number of non-boundary vertices in the n-gon at which k polygons meet.

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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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A368756 Number of vertices in the hyperoctahedral (or cocktail party) graph of order n.

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A146213 Number of intersection points outside the n-gon of all lines through pairs of vertices of a regular n-gon.

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A332418 The number of vertices on a decagon 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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A332428 The number of vertices on a nonagon 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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