A092866 -id:A092866 - cp's OEIS frontend

A341878 The number of vertices on a vesica piscis formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

2, 5, 19, 65, 203, 429, 963, 1643, 2929, 4173, 7179, 9985, 14747, 19503, 27043, 34511, 46011, 57171, 73339, 87509, 111219, 132809, 162459, 190703, 229419, 266285, 315041, 361179, 423067, 480201, 556451, 627039, 718923, 805217, 915091, 1017379, 1148595, 1270569, 1423907, 1564359

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Feb 22 2021

nonn

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

Scott R. Shannon, Vertices for n = 4.
Scott R. Shannon, Vertices for n = 5.
Scott R. Shannon, Vertices for n = 10.
Scott R. Shannon, Vertices for n = 15.
Scott R. Shannon, Vertices for n = 16.
Wikipedia, Vesica piscis.

Cf. A341877 (regions), A342152 (edges), A342153 (n-gons), A007569, A092866, A340644, A340686.

A366485 Place n equally spaced points on each side of an equilateral triangle, and join each of these points by a chord to the 2*n new points on the other two sides: sequence gives number of edges in the resulting planar graph.

Original entry on oeis.org

3, 9, 48, 237, 684, 1962, 3630, 7617, 12654, 21114, 31170, 50280, 66687, 99342, 132756, 174567, 222495, 302553, 367158, 479226, 579057, 705432, 846477, 1055679, 1217541, 1460205, 1715088, 2011161, 2289753, 2729301, 3044637, 3561606, 4037604, 4587153, 5175597, 5865729, 6432138, 7327737

Offset: 0

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

nonn

See A366483 for further information. See A366483 and A366486 for images of the triangle.

Cf. A366483 (vertices), A366484 (interior vertices), A366486 (regions).
If the 3*n points are placed "in general position" instead of uniformly, we get sequences A366478, A365929, A366932, A367015.
If the 3*n points are placed uniformly and we also draw chords from the three corner points of the triangle to these 3*n points, we get A274585, A092866, A274586, A092867.

a(n) = A366483(n) + A366486(n) - 1 (Euler).

A333136 The number of vertices formed on a triangle with leg lengths equal to the Pythagorean triples by the straight line segments mutually connecting all vertices and all points that divide the sides into unit length parts.

Original entry on oeis.org

230, 5138, 13181, 29277, 48107, 100003, 173261, 256910, 247940, 541752, 554717, 869197, 1051503, 987045, 1333241, 1190131, 1843049, 2991447, 3073340, 4382249, 4630456, 4635744, 5914142, 6877208

Offset: 1

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

See A332978 for the Pythagorean triple ordering and the links for images of the triangles.

Cf. A332978 (regions), A333135 (n-gons), A333137 (edges), A103605 (Pythagorean triple ordering), A092866, A332599, A007569.

a(8)-a(24) from Lars Blomberg, Jun 07 2020

A338003 The number of vertices in a 4-pointed star 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

37, 653, 5517, 17153, 50349, 97037, 204329, 330613, 571021, 835713, 1298533, 1764125

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Oct 06 2020

See A338002 for a definition of the star and images for the number of regions.

Scott R. Shannon, Image of the star showing the vertex counts for n=1.
Scott R. Shannon, Image of the star showing the vertex counts for n=2.
Scott R. Shannon, Image of the star showing the vertex counts for n=3.
Scott R. Shannon, Image of the star showing the vertex counts for n=4.
Scott R. Shannon, Image of the star showing the vertex counts for n=5.

Cf. A338002 (number of regions), A092866, A332599, A007569, A333117, A333116.

a(8)-a(12) from Lars Blomberg, Apr 08 2021

A366479 Irregular triangle T(n,k) (n >= 0, k >= 2) read by rows. Consider the planar graph formed from an equilateral triangle with n equally spaced points placed on each edge, as discussed in A092867(n+1). Then T(n,k) is the number of interior points where exactly k chords cross.

Original entry on oeis.org

0, 3, 1, 42, 7, 123, 37, 6, 444, 90, 6, 3, 1053, 138, 33, 12, 1, 2145, 285, 63, 15, 0, 3, 4173, 481, 72, 24, 12, 6774, 790, 165, 30, 9, 3, 6, 10698, 1270, 183, 75, 24, 6, 6, 16827, 1584, 393, 102, 6, 12, 6, 3, 25746, 2220, 339, 135, 40, 12, 6, 6, 35052, 3084, 684, 177, 42, 18, 6, 9, 6

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Nov 07 2023

			Triangle begins:
0;
3, 1;
42, 7;
123, 37, 6;
444, 90, 6, 3;
1053, 138, 33, 12, 1;
2145, 285, 63, 15, 0, 3;
4173, 481, 72, 24, 12;
6774, 790, 165, 30, 9, 3, 6;
10698, 1270, 183, 75, 24, 6, 6;
16827, 1584, 393, 102, 6, 12, 6, 3;
25746, 2220, 339, 135, 40, 12, 6, 6;
35052, 3084, 684, 177, 42, 18, 6, 9, 6;
51378, 3667, 657, 237, 87, 30, 3, 0, 6;
67287, 5101, 1080, 255, 96, 21, 18, 15, 6;
87183, 6943, 1206, 393, 117, 57, 36, 24, 0, 0, 3;
113682, 8478, 1761, 486, 150, 27, 24, 30, 0, 15;
152460, 9927, 1557, 522, 180, 33, 51, 18, 12, 0, 1;
187131, 12585, 2559, 678, 180, 54, 24, 3, 24, 15, 6;
240942, 14190, 2358, 690, 318, 54, 42, 25, 12, 0, 15;
288459, 17866, 3372, 990, 342, 75, 48, 9, 36, 30, 0, 6;
...

Scott R. Shannon, Image for n = 1.
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 = 10.

Cf. A092867.

See also A092866 (row sums), A366480 (T(n,2)/3).

cp's OEIS Frontend

A341878 The number of vertices on a vesica piscis 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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A366485 Place n equally spaced points on each side of an equilateral triangle, and join each of these points by a chord to the 2*n new points on the other two sides: sequence gives number of edges in the resulting planar graph.

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A333136 The number of vertices formed on a triangle with leg lengths equal to the Pythagorean triples by the straight line segments mutually connecting all vertices and all points that divide the sides into unit length parts.

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A338003 The number of vertices in a 4-pointed star 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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A366479 Irregular triangle T(n,k) (n >= 0, k >= 2) read by rows. Consider the planar graph formed from an equilateral triangle with n equally spaced points placed on each edge, as discussed in A092867(n+1). Then T(n,k) is the number of interior points where exactly k chords cross.

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