cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

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

Keywords

Comments

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.

Links

Crossrefs

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.

Formula

a(n) = A366485(n) - A366486(n) + 1 (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

Views

Author

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

Keywords

Comments

See A366483 for further information.

Links

Crossrefs

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.

Formula

a(n) = A366485(n) - A366483(n) + 1 (Euler).

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

Views

Author

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

Keywords

Comments

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

Crossrefs

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.

Formula

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

A367303 Table read by antidiagonals: Place k equally spaced points on each side of a regular n-gon and join every pair of these n*k points by a chord; T(n,k) (n >= 3, k >= 0) gives the number of internal vertices in the resulting planar graph.

Original entry on oeis.org

0, 0, 0, 13, 1, 0, 96, 57, 5, 0, 285, 329, 200, 13, 0, 901, 1317, 1265, 379, 35, 0, 1605, 2861, 4410, 2437, 987, 49, 0, 3534, 7417, 11280, 7843, 5789, 1577, 126, 0, 5797, 12801, 24455, 21235, 19642, 9193, 3042, 161, 0, 9813, 23329, 46570, 46141, 49931, 33161, 17280, 4371, 330, 0
Offset: 3

Views

Author

Scott R. Shannon, Nov 13 2023

Keywords

Comments

See A366484 and A367277 for other images of the n-gons.

Examples

			The table begins:
0, 0, 13, 96, 285, 901, 1605, 3534, 5797, 9813, 14319, 23809, 30912, 47154, ...
0, 1, 57, 329, 1317, 2861, 7417, 12801, 23329, 36549, 64625, 80085, 138261, ...
0, 5, 200, 1265, 4410, 11280, 24455, 46570, 80960, 131510, 202555, 298630, ...
0, 13, 379, 2437, 7843, 21235, 46141, 87427, 150391, 249925, 388819, 569767, ...
0, 35, 987, 5789, 19642, 49931, 106120, 200508, 346619, 560602, 861252, ...
0, 49, 1577, 9193, 33161, 83313, 182649, 341369, 597097, 961681, 1490601, ...
0, 126, 3042, 17280, 57510, 145008, 306621, 576711, 994149, 1605267, 2462013, ...
0, 161, 4371, 25161, 87251, 218151, 469331, 877211, 1522141, 2452131, 3781431, ...
0, 330, 7293, 40524, 133738, 335126, 706310, 1324763, 2279695, 3676321, ...
0, 301, 7861, 53077, 182653, 456181, 990145, 1849453, 3207217, 5171377, ...
0, 715, 14924, 81705, 267930, 668733, 1406275, 2632604, 4524793, ...
0, 757, 19447, 107101, 360319, 893033, 1896567, 3536275, 6103763, ...
0, 1365, 27375, 148275, 483930, 1204305, 2528340, 4726650, 8116515, ...
.
.
.

Links

Crossrefs

Cf. A367302 (vertices), A367304 (regions), A367305 (edges), A366484 (first row), A367277 (second row).

Formula

a(n,k) = A367305(n,k) - A367304(n,k) + 1 - n*(k+1) (Euler).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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