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.

A329714 Irregular table read by rows: Take a heptagon with all diagonals drawn, as in A329713. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

Original entry on oeis.org

35, 7, 7, 0, 1, 504, 224, 112, 28, 2331, 1883, 1008, 273, 92, 7, 7658, 6314, 3416, 798, 182, 28, 18662, 17514, 8463, 2898, 714, 175, 28, 7, 0, 0, 0, 1, 40404, 35462, 18508, 5796, 1330, 266, 28
Offset: 1

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Author

Keywords

Comments

See the links in A329713 for images of the heptagons.

Examples

			A heptagon with no other points along its edges, n = 1, contains 35 triangles, 7 quadrilaterals, 7 pentagons, 1 heptagon and no other n-gons, so the first row is [35,7,7,0,1]. A heptagon with 1 point dividing its edges, n = 2, contains 504 triangles, 224 quadrilaterals, 112 pentagons, 28 hexagons and no other n-gons, so the second row is [504,224,112,28].
Triangle begins:
35, 7, 7, 0, 1;
504, 224, 112, 28;
2331, 1883, 1008, 273, 92, 7;
7658, 6314, 3416, 798, 182, 28;
18662, 17514, 8463, 2898, 714, 175, 28, 7, 0, 0, 0, 1;
40404, 35462, 18508, 5796, 1330, 266, 28;
73248, 71596, 35777, 11669, 2654, 651, 70, 49;
The row sums are A329713.
		

Crossrefs

Cf. A329713 (regions), A333112 (edges), A333113 (vertices), A331906, A007678, A092867, A331452.

A333112 The number of edges inside a heptagon 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

91, 1575, 10962, 35812, 96257, 201054, 389991, 668458, 1096508, 1675835, 2494989, 3536876, 4930408, 6639913, 8816458, 11425631, 14659085, 18433975, 23007579, 28257418, 34478871, 41557817, 49822388, 59079475, 69756253, 81641812, 95165210
Offset: 1

Views

Author

Keywords

Comments

See the links in A329713 for images of the heptagons.

Crossrefs

Cf. A329713 (regions), A329714 (n-gons), A333113 (vertices), A330845, A274586 , A332600, A331765.

Extensions

a(8)-a(27) from Lars Blomberg, May 13 2020

A333113 The number of vertices inside a heptagon 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

42, 708, 5369, 17417, 47796, 99261, 194278, 331955, 546805, 833946, 1245314, 1762265, 2461837, 3311680, 4402405, 5700598, 7322231, 9200878, 11494161, 14108123, 17224438, 20752264, 24894009, 29506128, 34854099, 40780391, 47552050
Offset: 1

Views

Author

Keywords

Comments

See the links in A329713 for images of the heptagons.

Crossrefs

Cf. A329713 (regions), A329714 (n-gons), A333112 (edges), A330846, A092866, A332599, A007569.

Extensions

a(8)-a(27) from Lars Blomberg, May 13 2020

A335757 a(n) is the number of regions formed by n-secting the angles of a heptagon.

Original entry on oeis.org

1, 14, 78, 112, 50, 252, 484, 532, 848, 448, 1261, 1316, 1751, 1862, 1590, 2436, 3053, 3136, 3872, 2912, 4607, 4844, 5629, 5824, 5335, 6734, 7813, 7966, 9073, 7056, 10186, 10500, 11712, 11984, 11299, 13258, 14701, 14980, 16486, 14406, 17998, 18312, 19972
Offset: 1

Views

Author

Lars Blomberg, Jun 22 2020

Keywords

Crossrefs

Cf. A329713 (n-sected sides, not angles), A335758 (vertices), A335759 (edges), A335760 (ngons).

A367323 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 regions in the resulting planar graph.

Original entry on oeis.org

1, 12, 4, 75, 56, 11, 252, 340, 170, 24, 715, 1120, 1161, 408, 50, 1572, 3264, 3900, 2268, 868, 80, 3109, 6264, 10741, 8208, 5594, 1488, 154, 5676, 13968, 22380, 20832, 18396, 9312, 2754, 220, 9291, 22904, 44491, 44640, 48462, 31552, 16858, 4220, 375
Offset: 3

Views

Author

Keywords

Comments

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

Examples

			The table begins:
1, 12, 75, 252, 715, 1572, 3109, 5676, 9291, 14556, 22081, 32502, 44935, 62868, ...
4, 56, 340, 1120, 3264, 6264, 13968, 22904, 38748, 58256, 95656, 120960, ...
11, 170, 1161, 3900, 10741, 22380, 44491, 76610, 126336, 194070, 290651, ...
24, 408, 2268, 8208, 20832, 44640, 89214, 154752, 249906, 390012, 590658, ...
50, 868, 5594, 18396, 48462, 101794, 195714, 336504, 549704, 841890, 1249676, ...
80, 1488, 9312, 31552, 83432, 174816, 339816, 584176, 953416, 1463936, 2173976, ...
154, 2754, 16858, 55098, 142318, 298350, 568162, 975294, 1585666, 2426292, ...
220, 4220, 25220, 84280, 217800, 456640, 873090, 1501520, 2436020, 3736540, ...
375, 6732, 39887, 129492, 330903, 692648, 1311443, 2248840, 3645885, 5574756, ...
444, 9000, 52056, 178200, 462504, 963576, 1854432, 3180816, 5157612, 7906080, ...
781, 13962, 80783, 261222, 662663, 1385332, 2613521, 4478188, 7246331, ...
952, 18676, 107142, 352828, 891870, 1870876, 3525494, 6053768, 9778370, ...
1456, 25860, 146956, 474000, 1196116, 2498010, 4700776, 8050080, 13008106, ...
1696, 33152, 188000, 615328, 1547792, 3244000, 6095600, 10458560, 16876160, ...
2500, 44098, 247334, 795634, 1999762, 4173296, 7838004, 13416740, ...
2466, 52236, 302148, 991800, 2502000, 5229396, 9846234, ...
4029, 70604, 391781, 1258028, 3152101, 6574000, 12328417, ...
4500, 86240, 475800, 1546280, 3865240, 8085040, 15143880, ...
6175, 107562, 591655, 1897182, 4741633, 9883986, ...
6820, 129448, 706288, 2288880, 5703698, 11924132, ...
9086, 157412, 859718, 2753192, 6866858, 14307932, ...
9024, 181152, 1001640, 3254160, 8117304, ...
12926, 222850, 1209776, 3870250, 9636276, ...
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Crossrefs

Cf. A367322 (vertices), A367324 (edges), A092867 (1st row), A255011 (2nd row), A331929 (3rd row), A331931 (4th row), A329713 (5th row), A333075 (6th row), A332421 (7th row), A333139 (8th row), A007678 (1st column).

Formula

T(n,k) = A367324(n,k) - A367322(n,k) + 1 (Euler).
Showing 1-5 of 5 results.