A092866 -id:A092866 - cp's OEIS frontend

A357007 Number of vertices in an equilateral triangle when n internal equilateral triangles are drawn between the 3n points that divide each side into n+1 equal parts.

3, 6, 15, 30, 51, 66, 111, 150, 171, 246, 303, 312, 435, 510, 543, 678, 771, 765, 975, 1059, 1131, 1326, 1455, 1488, 1731, 1878, 1899, 2178, 2355, 2376, 2703, 2886, 2955, 3270, 3444, 3420, 3891, 4110, 4191, 4485, 4803, 4878, 5295, 5526, 5544, 6078, 6351, 6396, 6915, 7206, 7311, 7794, 8115

Offset: 0

Scott R. Shannon, Sep 08 2022

nonn

See A356984 for further images.

Scott R. Shannon, Table of n, a(n) for n = 0..250
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 = 5. This is the first term that forms intersections with non-simple vertices.
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 50.
Scott R. Shannon, Image for n = 100.
Scott R. Shannon, Image for n = 200.

Cf. A356984 (regions), A357008 (edges), A092866, A091908, A333026, A344657.

a(n) = A357008(n) - A356984(n) + 1 by Euler's formula.

Conjecture: a(n) = 3*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.

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

A333109 The number of vertices 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

57, 1145, 8417, 29121, 80345, 167105, 333297, 570969, 939113, 1441153, 2153937, 3029913, 4262929, 5741473, 7606745, 9876585, 12690553, 15921777, 19922289, 24430633, 29834073, 35990065, 43151521, 51068689

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), A333110 (edges), A330846, A092866, A332599, A007569.

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

A333113 The number of vertices 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

42, 708, 5369, 17417, 47796, 99261, 194278, 331955, 546805, 833946, 1245314, 1762265, 2461837, 3311680, 4402405, 5700598, 7322231, 9200878, 11494161, 14108123, 17224438, 20752264, 24894009, 29506128, 34854099, 40780391, 47552050

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), A333112 (edges), A330846, A092866, A332599, A007569.

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).

A333116 The number of vertices inside a hexagram 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

109, 2761, 16387, 63943, 167071, 357919, 711895, 1283419, 2040187, 3173851, 4909351, 6730795, 9868711, 13101883, 16984963, 23055523, 29896135, 36496711, 47223703, 56703265, 68999605, 84927301, 103692535, 119208667

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 08 2020

See the links in A331908 for images of the hexagrams.

Index entries for sequences related to stained glass windows

Cf. A331908 (regions), A331909 (n-gons), A333049 (edges), A092866, A332599, A007569.

a(6)-a(24) from Lars Blomberg, May 10 2020

A333117 The number of vertices inside a pentagram 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

26, 866, 6771, 24221, 64306, 132701, 259761, 453016, 734131, 1134081, 1673056, 2384606, 3326391, 4478286, 5941196, 7710796, 9901136, 12407581, 15497721, 19088991, 23256266, 28021386, 33537586, 39846196, 47092241, 55136771, 64103776, 74213991, 85642556, 98039461

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 08 2020

nonn

See the links in A331906 for images of the pentagrams.

Cf. A331906 (regions), A331907 (n-gons), A333118 (edges), A092866, A332599, A007569.

a(7)-a(30) from Lars Blomberg, May 06 2020

cp's OEIS Frontend

A357007 Number of vertices 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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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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A333109 The number of vertices 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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A333113 The number of vertices 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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A333116 The number of vertices inside a hexagram 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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A333117 The number of vertices inside a pentagram 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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