A340688 -id:A340688 - cp's OEIS frontend

A340685 The number of regions inside a concave circular triangle formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

1, 12, 28, 102, 187, 462, 868, 1590, 3046, 4398, 6541, 8646, 12256, 15336, 20947, 26610, 33478, 41832, 50821, 64710, 77110, 97878, 113932, 136560, 160849, 185220, 216286, 246450, 289945, 324594, 372976, 426936, 472231, 537366, 598606, 685650, 762736, 858546, 943684, 1043442, 1143751, 1258800

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 16 2021

nonn

The terms are from numeric computation - no formula for a(n) is currently known.

Scott R. Shannon, Regions for n = 2.
Scott R. Shannon, Regions for n = 3.
Scott R. Shannon, Regions for n = 4.
Scott R. Shannon, Regions for n = 6.
Scott R. Shannon, Regions for n = 8.
Scott R. Shannon, Regions for n = 10.
Scott R. Shannon, Regions for n = 14.
Scott R. Shannon, Regions for n = 15.
Wikipedia, Circular triangle.

Cf. A340686 (vertices), A340687 (edges), A340688 (n-gons), A340639, A007678, A092867.

A340686 The number of vertices on a concave circular triangle 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

3, 10, 24, 76, 168, 391, 819, 1447, 2949, 4153, 6393, 8293, 12048, 14857, 20670, 25972, 33123, 41026, 50379, 63700, 76560, 96628, 113262, 135076, 160050, 183484, 215355, 244435, 288861, 322276, 371742, 424279, 470838, 534376, 597033, 682264, 760959, 854755, 941706, 1039255, 1141569, 1254190

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 16 2021

nonn

The terms are from numeric computation - no formula for a(n) is currently known.

Scott R. Shannon, Vertices for n = 2.
Scott R. Shannon, Vertices for n = 3.
Scott R. Shannon, Vertices for n = 4.
Scott R. Shannon, Vertices for n = 6.
Scott R. Shannon, Vertices for n = 8.
Scott R. Shannon, Vertices for n = 10.
Scott R. Shannon, Vertices for n = 14.
Scott R. Shannon, Vertices for n = 15.
Wikipedia, Circular triangle.

Cf. A340685 (regions), A340687 (edges), A340688 (n-gons), A340644, A007678, A092867.

A340687 The number of edges on a concave circular triangle 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

3, 21, 51, 177, 354, 852, 1686, 3036, 5994, 8550, 12933, 16938, 24303, 30192, 41616, 52581, 66600, 82857, 101199, 128409, 153669, 194505, 227193, 271635, 320898, 368703, 431640, 490884, 578805, 646869, 744717, 851214, 943068, 1071741, 1195638, 1367913, 1523694, 1713300, 1885389, 2082696

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 16 2021

nonn

The terms are from numeric computation - no formula for a(n) is currently known.

See A340685 for images of the regions and A340686 for images of the vertices.

Wikipedia, Circular triangle.

Cf. A340685 (regions), A340686 (vertices), A340688 (n-gons), A340613, A007678, A092867.

A342153 Irregular table read by rows: Take a vesica piscis with all diagonals drawn, as in A341877. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

Original entry on oeis.org

0, 4, 18, 6, 52, 28, 4, 120, 78, 34, 4, 252, 188, 56, 12, 470, 348, 184, 40, 4, 2, 808, 648, 300, 56, 8, 1282, 1118, 548, 138, 20, 4, 2036, 1772, 644, 156, 28, 8, 2878, 2804, 1252, 388, 96, 10, 4172, 4024, 1728, 468, 100, 28, 5752, 5682, 2600, 866, 162, 46, 7912, 7676, 3420, 1024, 196, 44, 16

Offset: 1

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

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.

			A vesica piscis with 1 point dividing its edges, n = 2, contains 4 triangles and no other n-gons, so the second row is [4]. A vesica piscis with 3 points dividing its edges, n = 4, contains 52 triangles, 28 quadrilaterals, 4 pentagons and no other n-gons, so the fourth row is [52, 28, 4].
The table begins:
0;
4;
18,6;
52,28,4;
120,78,34,4;
252,188,56,12;
470,348,184,40,4,2;
808,648,300,56,8;
1282,1118,548,138,20,4;
2036,1772,644,156,28,8;
2878,2804,1252,388,96,10;
4172,4024,1728,468,100,28;
5752,5682,2600,866,162,46;
7912,7676,3420,1024,196,44,16;
10388,10354,4868,1548,352,60,6;
13496,13808,6016,1836,388,80,4,0,4;
17310,17590,8376,2672,564,122,16,2;
22012,22364,10160,3152,712,124,20,4;
27440,27956,13162,4432,964,172,24,2,4;
33784,34736,15588,4640,1096,120,28;

Scott R. Shannon, Image of the k-gons for n=14.
Wikipedia, Vesica piscis.

Cf. A341877 (regions), A342152 (edges), A341878 (vertices), A331451, A331911, A340614, A340688.

cp's OEIS Frontend

A340685 The number of regions inside a concave circular triangle 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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A340686 The number of vertices on a concave circular triangle 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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A340687 The number of edges on a concave circular triangle 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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A342153 Irregular table read by rows: Take a vesica piscis with all diagonals drawn, as in A341877. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

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