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 1-3 of 3 results.

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).

A367304 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 regions in the resulting planar graph.

Original entry on oeis.org

1, 4, 1, 27, 8, 1, 130, 88, 16, 1, 385, 444, 246, 30, 1, 1044, 1544, 1376, 492, 57, 1, 2005, 3584, 4621, 2814, 1079, 88, 1, 4060, 8356, 11691, 9042, 6014, 1800, 163, 1, 6831, 14996, 25026, 23604, 20049, 10016, 3196, 230, 1, 11272, 26572, 47386, 50448, 50597, 34432, 17632, 4770, 386, 1
Offset: 3

Views

Author

Scott R. Shannon, Nov 13 2023

Keywords

Comments

See A367278 and A006533 for other images of the n-gons.

Examples

			The table begins:
1, 4, 27, 130, 385, 1044, 2005, 4060, 6831, 11272, 16819, 26436, 35737, 52147, ...
1, 8, 88, 444, 1544, 3584, 8356, 14996, 26572, 42144, 69988, 93264, 148364, ...
1, 16, 246, 1376, 4621, 11691, 25026, 47386, 82096, 133076, 204716, 301861, ...
1, 30, 492, 2814, 9042, 23604, 50448, 95244, 163890, 268848, 415146, 610476, ...
1, 57, 1079, 6014, 20049, 50597, 107171, 201916, 348559, 563375, 864977, ...
1, 88, 1800, 10016, 34432, 86360, 185856, 347976, 604248, 974184, 1502416, ...
1, 163, 3196, 17632, 58195, 146071, 308296, 578926, 997219, 1609453, 2467720, ...
1, 230, 4770, 26470, 89160, 222730, 474120, 887230, 1532880, 2470640, 3798120, ...
1, 386, 7525, 41053, 134729, 336678, 708753, 1327987, 2284151, 3682306, ...
1, 456, 9276, 56100, 187872, 468660, 1002300, 1873824, 3235104, 5214684, ...
1, 794, 15250, 82447, 269309, 670892, 1409630, 2637051, 4530891, ...
1, 966, 20286, 109956, 363552, 902174, 1904504, 3555020, 6119918, ...
1, 1471, 27811, 149266, 485761, 1207201, 2532751, 4732516, 8124511, ...
.
.
.

Links

Crossrefs

Cf. A367302 (vertices), A367303 (internal vertices), A367305 (edges), A366486 (first row), A367278 (second row), A006533 (second column).

Formula

a(n,k) = A367305(n,k) - A367302(n,k) + 1 (Euler).

A367305 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 edges in the resulting planar graph.

Original entry on oeis.org

3, 9, 4, 48, 16, 5, 237, 156, 30, 6, 684, 788, 460, 54, 7, 1962, 2880, 2660, 888, 105, 8, 3630, 6468, 9055, 5274, 2086, 152, 9, 7617, 15800, 23000, 16914, 11830, 3400, 306, 10, 12654, 27828, 49515, 44874, 39725, 19240, 6264, 410, 11, 21114, 49936, 93995, 96630, 100569, 67632, 34947, 9170, 737, 12
Offset: 3

Views

Author

Scott R. Shannon, Nov 13 2023

Keywords

Comments

See A367302, A367304, A366485, A367279 for images of the n-gons.

Examples

			The table begins:
3, 9, 48, 237, 684, 1962, 3630, 7617, 12654, 21114, 31170, 50280, 66687, 99342, ...
4, 16, 156, 788, 2880, 6468, 15800, 27828, 49936, 78732, 134656, 173396, ...
5, 30, 460, 2660, 9055, 23000, 49515, 93995, 163100, 264635, 407325, 600550, ...
6, 54, 888, 5274, 16914, 44874, 96630, 182718, 314334, 518832, 804030, 1180314, ...
7, 105, 2086, 11830, 39725, 100569, 213339, 402479, 695240, 1124046, 1726305, ...
8, 152, 3400, 19240, 67632, 169720, 368560, 689408, 1201416, 1935944, 2993104, ...
9, 306, 6264, 34947, 115749, 291132, 614979, 1155708, 1991448, 3214809, ...
10, 410, 9170, 51670, 176460, 440940, 943520, 1764520, 3055110, 4922870, ...
11, 737, 14850, 81620, 268521, 671869, 1415139, 2652837, 4563944, 7358736, ...
12, 780, 17172, 109224, 370584, 924912, 1992528, 3723372, 6442428, 10386180, ...
13, 1534, 30212, 164203, 537303, 1339702, 2815995, 5269758, 9055800, ...
14, 1750, 39774, 217112, 723940, 1795290, 3801168, 7091406, 12223806, ...
15, 2865, 55230, 297600, 969765, 2411595, 5061195, 9459285, 16241160, ...
.
.
.

Crossrefs

Cf. A367302 (vertices), A367303 (internal vertices), A367304 (regions), A366485 (first row), A367279 (second row).

Formula

a(n,k) = A367302(n,k) + A367304(n,k) - 1 (Euler).
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.