A274586 -id:A274586 - cp's OEIS frontend

A333031 Number of vertices in an equilateral triangle "frame" of size n (see Comments in A328526 for definition).

3, 10, 58, 183, 408, 777, 1323, 2142, 3276, 4773, 6717, 9264, 12507, 16554, 21351, 27090, 34047, 42318, 52008, 63453, 76566, 91371, 108249, 127608, 149487, 173982, 201072, 231225, 265002, 302487, 343857, 389856, 440175, 494670, 553611, 617610, 687477, 763320

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 05 2020

nonn

For n<=3 the terms equal A274585(n). See A328526 for images of the triangular frame.

Lars Blomberg, Table of n, a(n) for n = 1..65

Cf. A328526 (regions), A333030 (edges), A333032 (3-gons), A333033 (4-gons), A331776 (square frame), A274586 (filled triangle).

a(12) and beyond from Lars Blomberg, May 01 2020

A357008 Number of edges in an equilateral triangle when n internal equilateral triangles are drawn between the 3n points that divide each side into n+1 equal parts.

Original entry on oeis.org

3, 9, 27, 57, 99, 135, 219, 297, 351, 489, 603, 645, 867, 1017, 1107, 1353, 1539, 1575, 1947, 2127, 2295, 2649, 2907, 3021, 3459, 3753, 3855, 4359, 4707, 4821, 5403, 5769, 5967, 6537, 6897, 6957, 7779, 8217, 8451, 9003, 9603, 9837, 10587, 11061, 11211, 12153, 12699, 12897, 13827, 14409, 14715

Offset: 0

Scott R. Shannon, Sep 08 2022

nonn

See A356984 and A357007 for images of the triangles.

Scott R. Shannon, Table of n, a(n) for n = 0..250

Cf. A356984 (regions), A357007 (vertices), A274586, A332376, A333027, A344896.

a(n) = A356984(n) + A357007(n) - 1 by Euler's formula.

Conjecture: a(n) = 6*n^2 + 3 for equilateral triangles that only contain simple vertices when cut by n internal equilateral triangles. This is never the case if (n + 1) mod 3 = 0 for n > 3.

A332429 The number of edges 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

288, 5148, 33291, 108252, 283464, 591723, 1133928, 1941786, 3166605, 4837824, 7170120, 10164258, 14124447, 19017180, 25206777, 32659191, 41826366, 52595622, 65549880, 80507142, 98143200, 118271898, 141655707

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), A332428 (vertices), A330845, A274586, A332600, A331765.

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

A333030 Number of edges in an equilateral triangle "frame" of size n (see Comments in A328526 for definition).

Original entry on oeis.org

3, 21, 132, 432, 951, 1800, 3069, 4956, 7569, 11040, 15585, 21492, 28971, 38328, 49527, 62922, 79113, 98358, 120939, 147486, 177873, 212358, 251823, 296898, 347709, 404772, 468189, 538680, 617439, 704742, 801039, 907842, 1024629, 1151544, 1289319, 1438686

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 05 2020

nonn

For n<=3 the terms equal A274586(n). See A328526 for images of the triangular frame.

Lars Blomberg, Table of n, a(n) for n = 1..65

Cf. A328526 (regions), A333031 (vertices), A331776 (square frame), A274586 (filled triangle).

a(12) and beyond from Lars Blomberg, May 01 2020

A333036 Number of edges in an equal-armed cross with arms of length n (see Comments in A331456 for definition).

Original entry on oeis.org

8, 172, 964, 3316, 8524, 18188, 34540, 59908, 97324, 150028, 221692, 316124, 438364, 592364, 784060, 1019468, 1304996, 1644900, 2047412, 2519172, 3068556, 3704004, 4434044, 5265868, 6211652, 7276492, 8474484, 9813996, 11304292, 12958380, 14791124, 16810732

Offset: 0

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

nonn

See the links in A331456 for images of the crosses.

Lars Blomberg, Table of n, a(n) for n = 0..48

Cf. A331456 (regions), A333035 (vertices), A333037 (n-gons), A274586 , A332600, A331765.

a(11) and beyond from Lars Blomberg, May 30 2020

A333110 The number of edges on an octagon 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

136, 2632, 17728, 60672, 163776, 341920, 673112, 1155144, 1892528, 2905088, 4327912, 6104696, 8557008, 11532288, 15271624, 19829528, 25447640, 31957872, 39935984, 49008392, 59807600, 72151536, 86465832, 102403360

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 07 2020

See the links in A333075 for images of the octagons.

Cf. A333075 (regions), A333076 (n-gons), A333109 (vertices), A330845, A274586 , A332600, A331765.

a(7)-a(24) from Lars Blomberg, May 14 2020

A333112 The number of edges inside a heptagon 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

91, 1575, 10962, 35812, 96257, 201054, 389991, 668458, 1096508, 1675835, 2494989, 3536876, 4930408, 6639913, 8816458, 11425631, 14659085, 18433975, 23007579, 28257418, 34478871, 41557817, 49822388, 59079475, 69756253, 81641812, 95165210

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 07 2020

See the links in A329713 for images of the heptagons.

Cf. A329713 (regions), A329714 (n-gons), A333113 (vertices), A330845, A274586 , A332600, A331765.

a(8)-a(27) from Lars Blomberg, May 13 2020

A366483 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 vertices in the resulting planar graph.

Original entry on oeis.org

3, 6, 22, 108, 300, 919, 1626, 3558, 5824, 9843, 14352, 23845, 30951, 47196, 62773, 82488, 104544, 144784, 173694, 230008, 276388, 336927, 403452, 509218, 582417, 702228, 824956, 969387, 1098312, 1321978, 1463580, 1724190, 1952509, 2221497, 2505169, 2846908, 3103788, 3556143, 3978763, 4444003

Offset: 0

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

nonn

We start with the three corner points of the triangle, and add n further points along each edge. Including the corner points, we end up with n+2 points along each edge, and the edge is divided into n+1 line segments.

Each of the n points added to an edge is joined by 2*n chords to the points that were added to the other two edges. There are 3*n^2 chords.

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. A366484 (interior vertices), A366485 (edges), 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) = A366485(n) - A366486(n) + 1 (Euler).

A366484 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 interior vertices in the resulting planar graph.

Original entry on oeis.org

0, 0, 13, 96, 285, 901, 1605, 3534, 5797, 9813, 14319, 23809, 30912, 47154, 62728, 82440, 104493, 144730, 173637, 229948, 276325, 336861, 403383, 509146, 582342, 702150, 824875, 969303, 1098225, 1321888, 1463487, 1724094, 1952410, 2221395, 2505064, 2846800, 3103677, 3556029, 3978646, 4443883

Offset: 0

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

nonn

See A366483 for further information.

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. A366483 (vertices), A366485 (edges), 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) = A366485(n) - A366486(n) - 3*n - 2 (Euler).

A366486 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 regions in the resulting planar graph.

Original entry on oeis.org

1, 4, 27, 130, 385, 1044, 2005, 4060, 6831, 11272, 16819, 26436, 35737, 52147, 69984, 92080, 117952, 157770, 193465, 249219, 302670, 368506, 443026, 546462, 635125, 757978, 890133, 1041775, 1191442, 1407324, 1581058, 1837417, 2085096, 2365657, 2670429, 3018822, 3328351, 3771595, 4213602

Offset: 0

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

nonn

See A366483 for further information.

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. A366483 (vertices), A366484 (interior vertices), A366485 (edges).
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) = A366485(n) - A366483(n) + 1 (Euler).

cp's OEIS Frontend

A333031 Number of vertices in an equilateral triangle "frame" of size n (see Comments in A328526 for definition).

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A357008 Number of edges in an equilateral triangle when n internal equilateral triangles are drawn between the 3n points that divide each side into n+1 equal parts.

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A332429 The number of edges 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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A333030 Number of edges in an equilateral triangle "frame" of size n (see Comments in A328526 for definition).

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A333036 Number of edges in an equal-armed cross with arms of length n (see Comments in A331456 for definition).

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A333110 The number of edges on an octagon 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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A333112 The number of edges inside a heptagon 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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A366483 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 vertices in the resulting planar graph.

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A366484 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 interior vertices in the resulting planar graph.

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A366486 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 regions in the resulting planar graph.

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