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.

Previous Showing 21-27 of 27 results.

A333032 Number of triangular regions in an equilateral triangular "frame" of size n (see Comments in A328526 for definition).

Original entry on oeis.org

1, 12, 48, 147, 291, 519, 879, 1383, 2079, 3039, 4359, 5991, 7983, 10515, 13731, 17571, 22131, 27543, 33951, 41271, 49599, 59355, 70779, 83499, 97611, 113763, 132243, 152619, 175035, 199707, 226827, 256443, 288747, 324603, 364395, 407139, 452979, 503319
Offset: 1

Views

Author

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

Keywords

Comments

See A328526 for images of the triangular frame.

Links

Crossrefs

Cf. A328526 (regions), A333030 (edges), A333031 (vertices), A333033 (4-gons), A331776 (square frame), A274586 (filled triangle).

Extensions

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

A333033 Number of quadrilateral regions in an equilateral triangular "frame" of size n (see Comments in A328526 for definition).

Original entry on oeis.org

0, 0, 24, 102, 252, 504, 867, 1431, 2214, 3228, 4509, 6237, 8481, 11259, 14445, 18261, 22935, 28497, 34980, 42762, 51708, 61632, 72795, 85791, 100611, 117027, 134874, 154836, 177402, 202548, 230355, 261543, 295707, 332271, 371313, 413937, 460881, 511551
Offset: 1

Views

Author

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

Keywords

Comments

See A328526 for images of the triangular frame.

Links

Crossrefs

Cf. A328526 (regions), A333030 (edges), A333031 (vertices), A333032 (3-gons), A331776 (square frame), A274586 (filled triangle).

Extensions

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

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).

A333049 The number of edges 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

276, 6348, 36810, 137802, 356538, 760134, 1494702, 2668458, 4254294, 6597690, 10105662, 13949346, 20222142, 26925654, 35032086, 47139258, 60947286, 74831682, 96101142, 115904808, 141052560, 172931484, 210293622, 243217686
Offset: 1

Views

Author

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

Keywords

Comments

See the links in A331908 for images of the hexagrams.

Crossrefs

Cf. A331908 (regions), A331909 (n-gons), A333116 (vertices),
A274586, A332600, A331765.

Extensions

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

A333118 The number of edges 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

65, 1965, 14100, 49760, 130235, 268900, 522770, 911425, 1474680, 2276820, 3354695, 4785575, 6665240, 8973795, 11903415, 15446945, 19825715, 24850460, 31025390, 38221130, 46557865, 56092005, 67123385, 79765335, 94249750, 110346520, 128289075, 148525930, 171374335, 196206590
Offset: 1

Views

Author

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

Keywords

Comments

See the links in A331906 for images of the pentagrams.

Crossrefs

Cf. A331906 (regions), A331907 (n-gons), A333117 (vertices), A274586, A332600, A331765.

Extensions

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

A333137 The number of edges formed on a triangle with leg lengths equal to the Pythagorean triples by the straight line segments mutually connecting all vertices and all points that divide the sides into unit length parts.

Original entry on oeis.org

500, 10883, 27220, 61570, 98657, 208739, 353922, 533442, 507744, 1100007, 1146403, 1771007, 2168628, 2002321, 2719907, 2413390, 3787444, 6140737, 6238486, 8906032, 9394871, 9495582, 11939407, 14063303
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Mar 09 2020

Keywords

Comments

See A332978 for the Pythagorean triple ordering and the links for images of the triangles.

Crossrefs

Cf. A332978 (regions), A333135 (n-gons), A333136 (vertices), A103605 (Pythagorean triple ordering), A274586 , A332600, A331765.

Extensions

a(8)-a(24) from Lars Blomberg, Jun 07 2020

A367324 Table read by antidiagonals: Place k equally spaced points on each side of a regular n-gon and join every pair of the n*(k+1) boundary 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, 21, 8, 132, 92, 20, 429, 596, 290, 42, 1272, 1936, 2215, 708, 91, 2826, 6020, 7405, 4020, 1575, 136, 5640, 11088, 21150, 15120, 10962, 2632, 288, 10461, 26260, 43490, 38544, 35812, 17728, 5148, 390, 17094, 42144, 88230, 83136, 96257, 60672, 33291, 7800, 715
Offset: 3

Views

Author

Scott R. Shannon and N. J. A. Sloane, Nov 14 2023

Keywords

Comments

See A367322, A367323 and the cross references for images of the n-gons.

Examples

			The table begins:
3, 21, 132, 429, 1272, 2826, 5640, 10461, 17094, 26847, 41046, 61041, 84051, ...
8, 92, 596, 1936, 6020, 11088, 26260, 42144, 72296, 107832, 183340, 222940, ...
20, 290, 2215, 7405, 21150, 43490, 88230, 151135, 250825, 384360, 578840, ...
42, 708, 4020, 15120, 38544, 83136, 169686, 294678, 475500, 746340, 1140624, ...
91, 1575, 10962, 35812, 96257, 201054, 389991, 668458, 1096508, 1675835, ...
136, 2632, 17728, 60672, 163776, 341920, 673112, 1155144, 1892528, 2905088, ...
288, 5148, 33291, 108252, 283464, 591723, 1133928, 1941786, 3166605, 4837824, ...
390, 7800, 48870, 164470, 430840, 900890, 1735800, 2982660, 4849740, 7438490, ...
715, 12793, 79134, 255552, 660033, 1376870, 2619287, 4482654, 7284904, ...
756, 16512, 99348, 346140, 912960, 1894920, 3685056, 6313164, 10261200, ...
1508, 26806, 160641, 516932, 1322802, 2757339, 5221996, 8932664, 14483183, ...
1722, 35546, 210658, 696682, 1773828, 3718400, 7030464, 12067720, 19517596, ...
2835, 49995, 292590, 939720, 2388825, 4976130, 9394815, 16064970, 26003640, ...
3088, 63456, 370784, 1217664, 3081472, 6455872, 12162640, 20861328, 33700320, ...
4896, 85680, 493017, 1579436, 3995102, 8318525, 15667336, 26783636, ...
4320, 99036, 593784, 1958922, 4978872, 10395450, 19644408, ...
7923, 137693, 781470, 2499792, 6298633, 13109658, 24645983, ...
8360, 167160, 941940, 3068280, 7705420, 16112480, 30238400, ...
12180, 210378, 1180683, 3772692, 9476418, 19717089, ...
12782, 252296, 1400674, 4547884, 11375584, 23776236, ...
17963, 308591, 1716306, 5478232, 13725457, 28550084, ...
16344, 350448, 1981416, 6460080, 16185624, ...
25600, 437700, 2415825, 7704700, 19262750, ...
.
.
.

Crossrefs

Cf. A367322 (vertices), A367323 (regions), A274586 (1st row), A331448 (2nd row), A329710 (3rd row), A330845 (4th row), A333112 (5th row), A333110 (6th row), A332429 (7th row), A332419 (8th row), A135565 (1st column).

Formula

T(n,k) = A367322(n,k) + A367323(n,k) - 1 (Euler).
Previous Showing 21-27 of 27 results.

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

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