A135565 -id:A135565 - cp's OEIS frontend

A359118 Number of edges in the planar Farey Ring graph FR(n) defined in A359116, including the regions between the convex hull and the bounding circle.

1, 2, 12, 48, 457, 1027, 6190, 14652, 40852, 71601, 223637, 325661, 847984, 1269433, 2053303, 3157887, 6638971, 8490949, 16421392, 21323264, 30639928

Offset: 1

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

See A359116 and A359117 for images of the figure.

Cf. A359116 (vertices), A359117 (regions), A359119 (k-gons), A358888, A006842, A006843, A005728, A135565.

a(n) = A359116 + A359117 - 1 by Euler's formula.

A368815 Number of edges in a regular 2n-gon when all vertices are connect by straight lines except for the n lines joining diametrically opposite vertices.

Original entry on oeis.org

0, 4, 24, 104, 320, 648, 1540, 2832, 3996, 7860, 12100, 15552, 25376, 35084, 36840, 62752, 81600, 97272, 131556, 163720, 178332, 245564, 296516, 336960, 421600, 497172, 557280, 678552, 786016, 836520, 1038500, 1185344, 1301124, 1525444, 1720740, 1893600, 2166720, 2419612, 2628600, 2991440

Offset: 1

Scott R. Shannon, Jan 06 2024

nonn

See A368813 and A368814 for images of the 2n-gons.

Cf. A368813 (regions), A368814 (vertices), A368816 (k-gons), A368757, A135565.

a(n) = A368813(n) + A368814(n) - 1 by Euler's formula.

A350502 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of edges in a regular n-gon after k generations of mitosis.

Original entry on oeis.org

3, 3, 4, 3, 8, 5, 3, 8, 20, 6, 3, 8, 35, 42, 7, 3, 8, 50, 66, 91, 8, 3, 8, 65, 66, 308, 136, 9, 3, 8, 80, 66, 630, 232, 288, 10, 3, 8, 95, 66, 1057, 232, 1305, 390, 11, 3, 8, 110, 66, 1589, 232, 2808, 900, 715, 12, 3, 8, 125, 66, 2226, 232, 4581, 1050, 3399, 756, 13

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Jan 01 2022

See A350000 for further details and images of the n-gons.

			The table begins:
.
      |               Number of edges after k generations
  n\k |  0,     1,     2,      3,      4,      5,      6,      7,      8, ...
----------------------------------------------------------------------------------
   3  |  3,     3,     3,      3,      3,      3,      3,      3,      3, ...
   4  |  4,     8,     8,      8,      8,      8,      8,      8,      8, ...
   5  |  5,    20,    35,     50,     65,     80,     95,    110,    125, ...
   6  |  6,    42,    66,     66,     66,     66,     66,     66,     66, ...
   7  |  7,    91,   308,    630,   1057,   1589,   2226,   2968,   3815, ...
   8  |  8,   136,   232,    232,    232,    232,    232,    232,    232, ...
   9  |  9,   288,  1305,   2808,   4581,   6624,   8937,  11520,  14373, ...
  10  | 10,   390,   900,   1050,   1200,   1350,   1500,   1650,   1800, ...
  11  | 11,   715,  3399,   7271,  11803,  16995,  22847,  29359,  36531, ...
  12  | 12,   756,  1428,   1428,   1428,   1428,   1428,   1428,   1428, ...
  13  | 13,  1508,  9061,  22243,  38350,  57382,  79339, 104221, 132028, ...
  14  | 14,  1722,  4704,   6174,   7644,   9114,  10584,  12054,  13524, ...
  15  | 15,  2835, 15345,  35880,  62370,  94035, 130875, 172890, 220080, ...
  16  | 16,  3088,  9424,  12496,  14416,  16336,  18256,  20176,  22096, ...
  17  | 17,  4896, 30294,  77758, 141440, 220932, 316234, 427346, 554268, ...
  18  | 18,  4320, 11376,  16686,  21528,  25578,  29628,  33678,  37728, ...
  19  | 19,  7923, 48773, 122607, 218101, 335255, 474069, 634543, 816677, ...
  20  | 20,  8360, 30840,  48260,  60560,  72860,  85160,  97460, 109760, ...
  21  | 21, 12180, 66738, 153069, 260505, 389046, 538692, 709443, 901299, ...
  22  | 22, 12782, 49148,  79442, 100232, 121022, 141812, 162602, 183392, ...
.

Cf. A350000 (n-gons), A350501 (vertices), A135565 (column 1), A349968 (column 2), A331450, A349967.

A351129 Number of regions in a regular n-gon with all diagonals drawn whose edges all have the same number of facing edges.

Original entry on oeis.org

1, 0, 1, 0, 1, 8, 1, 0, 1, 132, 66, 56, 46, 144, 171, 576, 305, 620, 652, 616, 852, 1296, 1376, 1482, 1891, 1820, 2379, 4530, 3163, 3328, 3532, 4046, 4656, 4896, 6661, 6460, 7411, 7560, 9595, 11676, 10923, 13552, 10936, 13294, 14806, 17232, 17935, 17200, 20452, 20540, 24964, 27270

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Feb 02 2022

nonn

See A351045 for details of an edge's count of facing edges in an n-gon with all diagonals drawn.

			a(5) = 1. A pentagon with all diagonals drawn contains a central pentagon which is surrounded by five other triangles and therefore all its edges have a facing edge count of 6. See the attached image.
a(8) = 8. An octagon with all diagonals drawn contains eight central triangles all of which are surrounded by three other triangles and therefore all their edges have a facing edge count of 4. See the attached image.
a(15) = 46. A 15-gon with all diagonals drawn contains one central 15-gon which is surrounded by triangles, thirty quadrilaterals which are surrounded by other quadrilaterals, and fifteen triangles which are surrounded by pentagons. This gives a total of forty-six regions whose edges all have the same facing edge count. See the attached image.

Scott R. Shannon, Table of n, a(n) for n = 3..100
Scott R. Shannon, Image for n = 5. In this and other images the regions with edges with the same facing edge count are highlighted in the corresponding edge color.
Scott R. Shannon, Image for n = 8.
Scott R. Shannon, Image for n = 12.
Scott R. Shannon, Image for n = 15.
Scott R. Shannon, Image for n = 18.
Scott R. Shannon, Image for n = 24.

Cf. A351045, A135565, A007678, A342222, A349784, A344899, A342236.

A353991 The regular m-gons with all diagonals drawn that contain internal vertices with vertex-surrounding polygons with 4 sides. See A353876.

Original entry on oeis.org

4, 12, 24, 26, 36, 42, 48, 60, 72, 78, 84, 96, 106, 108, 120, 132, 144

Offset: 1

Scott R. Shannon, May 13 2022

See A353876 for further details. All terms are even as a regular polygon with an odd number of edges with all diagonals drawn has only simple interior vertices, i.e. all vertices are created by the crossing of only two lines. Each vertex that creates a 4-sided vertex-surrounding polygon is connected to two other edge vertices as well as the central vertex of this polygon. Such vertices must therefore be created by the crossing of at least three lines, so cannot be vertices of a regular polygon with an odd number of sides.

Scott R. Shannon, Image of the 12-gon. The vertex-surrounding polygons with 4 sides are highlighted in darkened red.

Cf. A353876, A007569, A007678, A135565, A351045.

a(13)-a(17) added by Scott R. Shannon, May 14 2022

A367190 Table read by antidiagonals: Place k points in general position on each side of a regular n-gon and join every pair of the n*(k+1) boundary points by a chord; T(n,k) (n >= 3, k >= 0) gives number of edges in the resulting planar graph.

Original entry on oeis.org

3, 24, 8, 153, 124, 20, 588, 780, 390, 42, 1635, 2816, 2370, 939, 91, 3708, 7480, 8300, 5568, 1932, 136, 7329, 16428, 21600, 19149, 11193, 3512, 288, 13128, 31724, 46770, 49242, 37996, 20176, 5994, 390, 21843, 55840, 89390, 105747, 96915, 67936, 33750, 9455, 715

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Nov 09 2023

"In general position" implies that the internal lines (or chords) formed from the n*k edge points only have simple intersections; there is no interior points where three or more such chords meet. Note that for even-n n-gons, with n>=6, the chords from the n corner points do create non-simple intersections.

See A367183 and A366253 for images of the n-gons.

			The table begins:
3, 24, 153, 588, 1635, 3708, 7329, 13128, 21843, 34320, 51513, 74484, 104403,...
8, 124, 780, 2816, 7480, 16428, 31724, 55840, 91656, 142460, 211948, 304224,...
20, 390, 2370, 8300, 21600, 46770, 89390, 156120, 254700, 393950, 583770,...
42, 939, 5568, 19149, 49242, 105747, 200904, 349293, 567834, 875787, 1294752,...
91, 1932, 11193, 37996, 96915, 206976, 391657, 678888, 1101051, 1694980,...
136, 3512, 20176, 67936, 172328, 366616, 691792, 1196576, 1937416, 2978488,...
288, 5994, 33750, 112716, 284580, 603558, 1136394, 1962360, 3173256, 4873410,...
390, 9455, 53040, 176325, 443750, 939015, 1765080, 3044165, 4917750, 7546575,...
715, 14432, 79761, 263692, 661595, 1397220, 2622697, 4518536, 7293627,...
756, 20712, 115008, 379476, 950340, 2004216, 3758112, 6469428, 10435956,...
1508, 29614, 161538, 530348, 1324960, 2790138, 5226494, 8990488, 14494428,...
1722, 40243, 220024, 721245, 1799434, 3785467, 7085568, 12181309, 19629610,...
2835, 54420, 293985, 960300, 2391675, 5025960, 9400545, 16152360, 26017875,...
3088, 70800, 383904, 1252960, 3117648, 6546768, 12238240, 21019104,...
.
.
.

Cf. A367119 (first row), A367122 (second row), A135565 (first column), A367183 (vertices), A366253 (regions).

T(n,k) = A367183(n,k) + A366253(n,k) - 1 by Euler's formula.
Conjectures:
T(3,k) = A367119(k) = (9/2)*k^4 + 6*k^3 + (9/2)*k^2 + 6*k + 3.
T(4,k) = A367122(k) = 17*k^4 + 38*k^3 + 37*k^2 + 24*k + 8.
T(5,k) = 45*k^4 + 120*k^3 + 130*k^2 + 75*k + 20.
T(6,k) = (195/2)*k^4 + 285*k^3 + (657/2)*k^2 + 186*k + 42.
T(7,k) = (371/2)*k^4 + 574*k^3 + (1379/2)*k^2 + 392*k + 91.
T(8,k) = 322*k^4 + 1036*k^3 + 1282*k^2 + 736*k + 136.
T(9,k) = 522*k^4 + 1728*k^3 + 2187*k^2 + 1269*k + 288.
T(10,k) = (1605/2)*k^4 + 2715*k^3 + (6995/2)*k^2 + 2050*k + 390.

A367324 Table read by antidiagonals: Place k equally spaced points on each side of a regular n-gon and join every pair of the n*(k+1) boundary points by a chord; T(n,k) (n >= 3, k >= 0) gives the number of edges in the resulting planar graph.

Original entry on oeis.org

3, 21, 8, 132, 92, 20, 429, 596, 290, 42, 1272, 1936, 2215, 708, 91, 2826, 6020, 7405, 4020, 1575, 136, 5640, 11088, 21150, 15120, 10962, 2632, 288, 10461, 26260, 43490, 38544, 35812, 17728, 5148, 390, 17094, 42144, 88230, 83136, 96257, 60672, 33291, 7800, 715

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Nov 14 2023

See A367322, A367323 and the cross references for images of the n-gons.

			The table begins:
3, 21, 132, 429, 1272, 2826, 5640, 10461, 17094, 26847, 41046, 61041, 84051, ...
8, 92, 596, 1936, 6020, 11088, 26260, 42144, 72296, 107832, 183340, 222940, ...
20, 290, 2215, 7405, 21150, 43490, 88230, 151135, 250825, 384360, 578840, ...
42, 708, 4020, 15120, 38544, 83136, 169686, 294678, 475500, 746340, 1140624, ...
91, 1575, 10962, 35812, 96257, 201054, 389991, 668458, 1096508, 1675835, ...
136, 2632, 17728, 60672, 163776, 341920, 673112, 1155144, 1892528, 2905088, ...
288, 5148, 33291, 108252, 283464, 591723, 1133928, 1941786, 3166605, 4837824, ...
390, 7800, 48870, 164470, 430840, 900890, 1735800, 2982660, 4849740, 7438490, ...
715, 12793, 79134, 255552, 660033, 1376870, 2619287, 4482654, 7284904, ...
756, 16512, 99348, 346140, 912960, 1894920, 3685056, 6313164, 10261200, ...
1508, 26806, 160641, 516932, 1322802, 2757339, 5221996, 8932664, 14483183, ...
1722, 35546, 210658, 696682, 1773828, 3718400, 7030464, 12067720, 19517596, ...
2835, 49995, 292590, 939720, 2388825, 4976130, 9394815, 16064970, 26003640, ...
3088, 63456, 370784, 1217664, 3081472, 6455872, 12162640, 20861328, 33700320, ...
4896, 85680, 493017, 1579436, 3995102, 8318525, 15667336, 26783636, ...
4320, 99036, 593784, 1958922, 4978872, 10395450, 19644408, ...
7923, 137693, 781470, 2499792, 6298633, 13109658, 24645983, ...
8360, 167160, 941940, 3068280, 7705420, 16112480, 30238400, ...
12180, 210378, 1180683, 3772692, 9476418, 19717089, ...
12782, 252296, 1400674, 4547884, 11375584, 23776236, ...
17963, 308591, 1716306, 5478232, 13725457, 28550084, ...
16344, 350448, 1981416, 6460080, 16185624, ...
25600, 437700, 2415825, 7704700, 19262750, ...
.
.
.

Cf. A367322 (vertices), A367323 (regions), A274586 (1st row), A331448 (2nd row), A329710 (3rd row), A330845 (4th row), A333112 (5th row), A333110 (6th row), A332429 (7th row), A332419 (8th row), A135565 (1st column).

T(n,k) = A367322(n,k) + A367323(n,k) - 1 (Euler).

A291949 Number of maximal cliques in the n-polygon diagonal intersection graph.

Original entry on oeis.org

1, 4, 10, 24, 49, 80, 162, 230, 451, 444, 884, 1036, 1845, 1840, 3026, 2412, 4921, 4960, 8022, 7854, 10925, 9864, 16150, 15860, 22545, 22204, 30218, 21300, 41261, 39776, 51678, 51102, 67025, 58680, 83990, 82802, 104715, 102480, 126362, 107184, 155445, 152020

Offset: 3

Eric W. Weisstein, Mar 09 2018

nonn

Maximal cliques in the n-polygon diagonal intersection graph are either triangles or edges that are not contained in a triangle. - Andrew Howroyd, Mar 11 2018

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

Cf. A062361, A135565, A300524.

Terms a(15) and beyond from Andrew Howroyd, Mar 11 2018

A300524 Number of (not necessarily maximal) cliques in the n-polygon diagonal intersection graph.

Original entry on oeis.org

8, 18, 41, 80, 169, 250, 514, 682, 1233, 1346, 2614, 2970, 4801, 5330, 8348, 7580, 13472, 14422, 20455, 21738, 30591, 28082, 43276, 45294, 59806, 62050, 80911, 66812, 106393, 110114, 139162, 142972, 177416, 171434, 223851, 228876, 278500

Offset: 3

Eric W. Weisstein, Mar 08 2018

nonn

Equal to 1 + vertex count + edge count + triangle count.

Eric Weisstein's World of Mathematics, Clique
Eric Weisstein's World of Mathematics, Polygon Diagonal Intersection Graph

Cf. A007569 (vertex count).
Cf. A062361 (edge count).
Cf. A135565 (triangle count).

a(n) = 1 + A007569(n) + A062361(n) + A135565(n).

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

A359118 Number of edges in the planar Farey Ring graph FR(n) defined in A359116, including the regions between the convex hull and the bounding circle.

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A368815 Number of edges in a regular 2n-gon when all vertices are connect by straight lines except for the n lines joining diametrically opposite vertices.

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A350502 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of edges in a regular n-gon after k generations of mitosis.

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A351129 Number of regions in a regular n-gon with all diagonals drawn whose edges all have the same number of facing edges.

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A353991 The regular m-gons with all diagonals drawn that contain internal vertices with vertex-surrounding polygons with 4 sides. See A353876.

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A367190 Table read by antidiagonals: Place k points in general position on each side of a regular n-gon and join every pair of the n*(k+1) boundary points by a chord; T(n,k) (n >= 3, k >= 0) gives number of edges in the resulting planar graph.

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A367324 Table read by antidiagonals: Place k equally spaced points on each side of a regular n-gon and join every pair of the n*(k+1) boundary points by a chord; T(n,k) (n >= 3, k >= 0) gives the number of edges in the resulting planar graph.

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A291949 Number of maximal cliques in the n-polygon diagonal intersection graph.

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A300524 Number of (not necessarily maximal) cliques in the n-polygon diagonal intersection graph.

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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.

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