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 1-5 of 5 results.

A333075 The number of regions inside 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

80, 1488, 9312, 31552, 83432, 174816, 339816, 584176, 953416, 1463936, 2173976, 3074784, 4294080, 5790816, 7664880, 9952944, 12757088, 16036096, 20013696, 24577760, 29973528, 36161472, 43314312, 51334672
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Mar 07 2020

Keywords

Comments

The terms are from numeric computation - no formula for a(n) is currently known.

Links

Crossrefs

Cf. A333076 (n-gons), A333109 (vertices), A333110 (edges), A007678, A092867, A331452, A331931.

Extensions

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

A333076 Irregular table read by rows: Take an octagon with all diagonals drawn, as in A333075. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

Original entry on oeis.org

56, 24, 800, 608, 64, 16, 4136, 3400, 1272, 464, 40, 13840, 10800, 5296, 1264, 288, 64, 33160, 30048, 14744, 4456, 840, 152, 32, 70832, 62208, 30848, 8656, 1936, 288, 48, 129624, 124224, 61560, 19312, 4168, 840, 64, 16, 0, 8, 225200, 210608, 107552, 32768
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Mar 07 2020

Keywords

Comments

See the links in A333075 for images of the octagons.

Examples

			An octagon with no other points along its edges, n = 1, contains 56 triangles, 24 quadrilaterals and no other n-gons, so the first row is [56,24]. An octagon with 1 point dividing its edges, n = 2, contains 800 triangles, 608 quadrilaterals, 64 pentagons, 16 hexagons and no other n-gons, so the second row is [800,608,64,16].
Table begins:
56,24;
800,608,64,16;
4136,3400,1272,464,40;
13840,10800,5296,1264,288,64;
33160,30048,14744,4456,840,152,32;
70832,62208,30848,8656,1936,288,48;
The rows sums are A333075.

Links

Crossrefs

Cf. A333075 (regions), A333109 (vertices), A333110 (edges), A331931, A331906, A007678, A092867, A331452.

Extensions

a(32) and beyond from Lars Blomberg, May 14 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

Views

Author

Scott R. Shannon and N. J. A. Sloane, Mar 07 2020

Keywords

Comments

See the links in A333075 for images of the octagons.

Crossrefs

Cf. A333075 (regions), A333076 (n-gons), A333110 (edges), A330846, A092866, A332599, A007569.

Extensions

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

A335771 a(n) is the number of edges formed by n-secting the angles of an octagon.

Original entry on oeis.org

8, 16, 48, 192, 648, 136, 1240, 1200, 1488, 2048, 3240, 1688, 4472, 4352, 4928, 5936, 7864, 4592, 9624, 9552, 10384, 11728, 14440, 10760, 16616, 16720, 17856, 19504, 23032, 18264, 25704, 25920, 27232, 29376, 33624, 26672, 36840, 36976, 38656, 41152, 46152
Offset: 1

Views

Author

Lars Blomberg, Jun 23 2020

Keywords

Comments

See A335769 for illustrations.

Links

Crossrefs

Cf. A333110 (n-sected sides, not angles), A335769 (regions), A335770 (vertices), A335772 (ngons).

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).
Showing 1-5 of 5 results.

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

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