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.

A367276 Place n equally spaced points on each side of a square, and join each of these points by a chord to the 3*n new points on the other three sides: sequence gives number of vertices in the resulting planar graph.

Original entry on oeis.org

4, 9, 69, 345, 1337, 2885, 7445, 12833, 23365, 36589, 64669, 80133, 138313, 176885, 233765, 312013, 455273, 513277, 741965, 819589, 1046245, 1310761, 1692961, 1772097, 2315289, 2713997, 3165125, 3552753, 4538845, 4602985, 6015561, 6432681, 7421345, 8550485, 9439621, 10063993, 12635769
Offset: 0

Views

Author

Scott R. Shannon, Nov 11 2023

Keywords

Comments

We start with the four corner points of the square, 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 3*n chords to the points that were added to the other three edges. There are 6*n^2 chords.

Links

Crossrefs

Cf. A367277 (interior vertices), A367278 (regions), A367279 (edges).
If the 4*n points are placed "in general position" instead of uniformly, we get sequences A334698, A367121, A367122.
If the 4*n points are placed uniformly and we also draw chords from the four corner points of the square to these 4*n points, we get A255011, A331448, A331449, A334690.

Formula

a(n) = A367279(n) - A367278(n) + 1 (Euler).

A367278 Place n equally spaced points on each side of a square, and join each of these points by a chord to the 3*n new points on the other three sides: sequence gives number of regions in the resulting planar graph.

Original entry on oeis.org

1, 8, 88, 444, 1544, 3584, 8356, 14996, 26572, 42144, 69988, 93264, 148364, 196120, 261536, 347544, 483056, 572832, 783252, 907472, 1141668, 1413512, 1775972, 1958684, 2466452, 2897004, 3384128, 3852068, 4733632, 5042324, 6263264, 6869784, 7878988, 9014840, 10035784, 10909592, 13109608
Offset: 0

Views

Author

Scott R. Shannon, Nov 11 2023

Keywords

Comments

See A367276 for further information.

Links

Crossrefs

Cf. A367276 (vertices), A367277 (interior vertices), A367279 (edges).

Formula

a(n) = A367279(n) - A367276(n) + 1 (Euler).

A367279 Place n equally spaced points on each side of a square, and join each of these points by a chord to the 3*n new points on the other three sides: sequence gives number of edges in the resulting planar graph.

Original entry on oeis.org

4, 16, 156, 788, 2880, 6468, 15800, 27828, 49936, 78732, 134656, 173396, 286676, 373004, 495300, 659556, 938328, 1086108, 1525216, 1727060, 2187912, 2724272, 3468932, 3730780, 4781740, 5611000, 6549252, 7404820, 9272476, 9645308, 12278824, 13302464, 15300332, 17565324, 19475404, 20973584
Offset: 0

Views

Author

Scott R. Shannon, Nov 11 2023

Keywords

Comments

See A367276 for further information. See A367276 and A367278 for images of the square.

Crossrefs

Cf. A367276 (vertices), A367277 (interior vertices), A367278 (regions).

Formula

a(n) = A367276(n) + A367278(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, ...
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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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