A341764 -id:A341764 - 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).

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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