A331929 -id:A331929 - cp's OEIS frontend

A341688 The number of regions inside a 2 by 1 ellipse formed by the straight line segments mutually connecting all points formed by dividing the ellipse into 2n equal angle sectors from its origin.

0, 4, 24, 84, 232, 524, 1052, 1868, 3144, 4876, 7440, 10724, 15124, 20604, 27632, 36124, 46672, 59108, 74184, 91488, 112380, 136044, 163724, 194924, 230932, 271124, 316992, 367748, 425124, 488116, 558820, 635964, 721824, 815044, 918132, 1029524, 1152012, 1283788, 1427964, 1582328, 1750760

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Feb 17 2021

nonn

The ellipse, with width twice its height, has points at its x-axis extrema and n-1 points both above and below the x-axis, 2n points in total. These are placed on the ellipse's perimeter by dividing it from the origin into 2n sectors of equal angle.

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

Scott R. Shannon, Regions for n = 3.
Scott R. Shannon, Regions for n = 5.
Scott R. Shannon, Regions for n = 9.
Scott R. Shannon, Regions for n = 19.
Scott R. Shannon, Regions for n = 24.
Scott R. Shannon, Regions for n = 13 using random distance-based coloring.
Wikipedia, Ellipse.

Cf. A341762 (vertices), A341764 (edges), A341800 (n-gons), A007678, A092867, A255011, A331929, A331931, A333075.

A341762 The number of vertices on a 2 by 1 ellipse formed by the straight line segments mutually connecting all points formed by dividing the ellipse into 2n equal angle sectors from its origin.

Original entry on oeis.org

2, 5, 19, 65, 195, 461, 971, 1737, 2995, 4617, 7203, 10385, 14779, 20125, 27155, 35481, 46051, 58277, 73395, 90323, 111403, 134765, 162539, 193385, 229515, 269301, 315331, 365617, 423195, 485617, 556603, 633145, 719299, 811845, 915275, 1025921, 1148811, 1279757, 1424395, 1577723, 1746803

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Feb 19 2021

nonn

See A341688 for a description of the ellipse.
Curiously the only ellipses found that have vertices with three or more lines crossing that are not on the x or y axes are those with a number of vertices equal to a multiple of ten.
The terms are from numeric computation - no formula for a(n) is currently known.

Scott R. Shannon, Image of the vertices for n = 3.
Scott R. Shannon, Image of the vertices for n = 5. Notice the off-axis vertices that have three lines crossing.
Scott R. Shannon, Image of the vertices for n = 9.
Scott R. Shannon, Image of the vertices for n = 10. Notice the off-axis vertices that have three or four lines crossing.
Scott R. Shannon, Image of the vertices for n = 20. Notice the off-axis vertices that have three or more lines crossing.
Scott R. Shannon, Image of the vertices for n = 21.
Wikipedia, Ellipse.

Cf. A341688 (regions), A341764 (edges), A341800 (n-gons), A007678, A092867, A255011, A331929, A331931, A333075.

A341800 Irregular table read by rows: Take a 2 by 1 ellipse with all diagonals drawn, as in A341688. Then T(n,k) = number of k-sided polygons in the figure containing 2n vertices, for k >= 3.

Original entry on oeis.org

0, 4, 18, 6, 52, 28, 4, 120, 84, 22, 6, 244, 192, 72, 16, 434, 432, 124, 54, 8, 748, 748, 300, 52, 16, 4, 1234, 1232, 482, 164, 26, 2, 4, 1896, 1940, 776, 220, 36, 8, 2764, 2926, 1332, 330, 78, 10, 3892, 4460, 1716, 536, 88, 28, 0, 4, 5580, 5918, 2642, 784, 152, 44, 4

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Feb 20 2021

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

See A341688 for a description of the ellipse and images of the regions, and A341762 for images of the vertices.

			A 2 by 1 ellipse consisting of 12 vertices, n = 6, contains 244 triangle, 192 quadrilaterals, 72 pentagons, 16 hexagons and no other n-gons, so the sixth row is [244, 192, 72, 16]. See the linked image.
The table begins:
0;
4;
18, 6;
52, 28, 4;
120, 84, 22, 6;
244, 192, 72, 16;
434, 432, 124, 54, 8;
748, 748, 300, 52, 16, 4;
1234, 1232, 482, 164, 26, 2, 4;
1896, 1940, 776, 220, 36, 8;
2764, 2926, 1332, 330, 78, 10;
3892, 4460, 1716, 536, 88, 28, 0, 4;
5580, 5918, 2642, 784, 152, 44, 4;
7508, 8204, 3540, 1108, 224, 12, 4, 4;
9902, 11202, 4636, 1472, 362, 44, 10, 4;
12984, 14508, 6208, 1920, 412, 80, 12;
16804, 18396, 8272, 2522, 522, 136, 20;
21212, 23352, 10580, 3144, 672, 112, 36;
26602, 28938, 13438, 4264, 766, 162, 12, 2;
32732, 36200, 16124, 5276, 952, 192, 12;
40026, 44216, 20038, 6564, 1302, 216, 16, 2;

Scott R. Shannon, Image of the n-gons for n = 6.
Wikipedia, Ellipse.

Cf. A341688 (regions), A341762 (vertices), A341764 (edges), A007678, A092867, A255011, A331929, A331931, A333075.

Row sums = A341688(n).

A341764 The number of edges on a 2 by 1 ellipse formed by the straight line segments mutually connecting all points formed by dividing the ellipse into 2n equal angle sectors from its origin.

Original entry on oeis.org

1, 8, 42, 148, 426, 984, 2022, 3604, 6138, 9492, 14642, 21108, 29902, 40728, 54786, 71604, 92722, 117384, 147578, 181810, 223782, 270808, 326262, 388308, 460446, 540424, 632322, 733364, 848318, 973732, 1115422, 1269108, 1441122, 1626888, 1833406, 2055444, 2300822, 2563544, 2852358, 3160050, 3497562

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Feb 19 2021

nonn

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

See A341688 for a description of the ellipse and images of the regions, and A341762 for images of the vertices.

Wikipedia, Ellipse.

Cf. A341688 (regions), A341762 (vertices), A341800 (n-gons), A007678, A092867, A255011, A331929, A331931, A333075.

a(n) = A341688(n) + A341762(n) - 1.

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

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

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

			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, ...
.
.
.

Scott R. Shannon, Image for T(12,2).
Scott R. Shannon, Image for T(23,1).

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).

T(n,k) = A367324(n,k) - A367322(n,k) + 1 (Euler).

A333643 Number of regions in a polygon whose boundary consists of n+2 equally spaced points around the arc of a semicircle. See Comments for precise definition.

Original entry on oeis.org

1, 4, 11, 25, 50, 91, 154, 234, 375, 550, 769, 1079, 1456, 1783, 2500, 3196, 3987, 5016, 6175, 7348, 9086, 10879, 12836, 15250, 17875, 20682, 24129, 27811, 31419, 36425, 41416, 46664, 52921, 59500, 66489, 74481, 82954, 91807, 102050, 112750, 123700, 136654

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 31 2020

nonn

A semicircular polygon with n+2 points is created by placing n+2 equally spaced vertices along a semicircle's arc, which includes the two end vertices. Now connect every pair of vertices by a straight line segment. The sequence gives the number of regions in the resulting figure.

Note that there is a curious relationship between the terms of this sequence and the number of regions in the 'general position' polygon given in A006522. They are a match except for every third term starting at a(8) = 234. Examining the images for n = 8,11,14,17 shows that these polygons have interior points at which three or more lines intersect, while the other n values have no such intersection points. Such multi-line intersection points will reduce the number of regions as compared to the general position polygon which has no multi-line intersection points. This is reflected by the terms in this sequence being lower than the corresponding value in A006522 for n = 8,11,14,... . Why every third value of n in this sequence starting at n = 8 leads to polygons having multiple line intersection points while other values of n do not is currently not known.

Lars Blomberg, Table of n, a(n) for n = 1..80
Scott R. Shannon, Illustration for n = 2.
Scott R. Shannon, Illustration for n = 3.
Scott R. Shannon, Illustration for n = 4.
Scott R. Shannon, Illustration for n = 5.
Scott R. Shannon, Illustration for n = 7.
Scott R. Shannon, Illustration for n = 10.
Scott R. Shannon, Illustration for n = 12.
Scott R. Shannon, Illustration for n = 15.
Scott R. Shannon, Illustration for n = 17.
Scott R. Shannon, Illustration for n = 19.
Scott R. Shannon, Illustration for n = 20.
Scott R. Shannon, Illustration for n = 10 with random distance-based coloring.
Scott R. Shannon, Illustration for n = 15 with random distance-based coloring.
Scott R. Shannon, Illustration for n = 19 with random distance-based coloring.
Scott R. Shannon, Illustration for n = 20 with random distance-based coloring.
Wikipedia, Semicircle.

Cf. A333642, A333519, A007678, A290865, A092867, A331452, A331929, A331931.

More terms from Lars Blomberg, Apr 20 2020

cp's OEIS Frontend

A341688 The number of regions inside a 2 by 1 ellipse formed by the straight line segments mutually connecting all points formed by dividing the ellipse into 2n equal angle sectors from its origin.

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A341762 The number of vertices on a 2 by 1 ellipse formed by the straight line segments mutually connecting all points formed by dividing the ellipse into 2n equal angle sectors from its origin.

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A341800 Irregular table read by rows: Take a 2 by 1 ellipse with all diagonals drawn, as in A341688. Then T(n,k) = number of k-sided polygons in the figure containing 2n vertices, for k >= 3.

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A341764 The number of edges on a 2 by 1 ellipse formed by the straight line segments mutually connecting all points formed by dividing the ellipse into 2n equal angle sectors from its origin.

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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.

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A333643 Number of regions in a polygon whose boundary consists of n+2 equally spaced points around the arc of a semicircle. See Comments for precise definition.

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