A255011 -id:A255011 - cp's OEIS frontend

A335353 Irregular table read by rows: Take a square and divide each of its sides into n equal parts giving a total of 4n nodes, draw straight line segments from node k to node (k+n+1) mod 4n, 0 <= k < 4*n. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

4, 16, 8, 0, 0, 0, 1, 32, 5, 32, 40, 8, 0, 0, 1, 64, 28, 16, 0, 0, 0, 0, 0, 0, 1, 80, 56, 24, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 96, 84, 24, 0, 0, 0, 0, 0, 0, 1, 128, 100, 40, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 144, 156, 32, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Lars Blomberg, Jun 04 2020

See A335350 for illustrations.

			Table begins:
4;
16, 8, 0, 0, 0, 1;
32, 5;
32, 40, 8, 0, 0, 1;
64, 28, 16, 0, 0, 0, 0, 0, 0, 1;
80, 56, 24, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
96, 84, 24, 0, 0, 0, 0, 0, 0, 1;
128, 100, 40, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
144, 156, 32, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
168, 188, 64, 16, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1;
200, 228, 40, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
248, 252, 88, 24, 8, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

Lars Blomberg, Table of n, a(n) for n = 1..10000

Cf. A335350 (regions), A335351 (edges), A335352 (vertices), A335354 (edges in central polygon), A255011, A335057, A335192.

A335354 a(n) is the number of edges in the central polygon formed in a square by dividing each of its sides into n equal parts giving a total of 4n nodes and drawing straight line segments from node k to node (k+n+1) mod 4n, 0 <= k < 4*n.

Original entry on oeis.org

0, 8, 4, 8, 12, 16, 12, 16, 20, 16, 20, 24, 28, 24, 28, 32, 28, 32, 36, 32, 36, 40, 44, 40, 44, 48, 44, 48, 52, 56, 52, 56, 60, 56, 60, 64, 68, 64, 68, 72, 68, 72, 76, 72, 76, 80, 84, 80, 84, 88, 84, 88, 92, 96, 92, 96, 100, 96, 100, 104, 100, 104, 108, 112

Offset: 1

Lars Blomberg, Jun 04 2020

nonn

For n=1 there is no central polygon.
The number of edges of the central polygon tends to grow as n increases, whereas for n = 16..500 the polygon with next-to-most edges has 8 of them.
See A335350 for illustrations.

Lars Blomberg, Table of n, a(n) for n = 1..500

Cf. A335350 (regions), A335351 (edges), A335352 (vertices), A335353 (n-gons), A255011, A335057, A335192.

A367121 Place n points in general position on each side of a square, and join every pair of the 4*n+4 boundary points by a chord; sequence gives number of regions in the resulting planar graph.

Original entry on oeis.org

4, 67, 406, 1441, 3796, 8299, 15982, 28081, 46036, 71491, 106294, 152497, 212356, 288331, 383086, 499489, 640612, 809731, 1010326, 1246081, 1520884, 1838827, 2204206, 2621521, 3095476, 3630979, 4233142, 4907281, 5658916, 6493771, 7417774, 8437057, 9557956, 10787011, 12130966

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Nov 05 2023

nonn

"In general position" implies that the internal lines (or chords) only have simple intersections. There is no interior point where three or more chords meet.

Note that although the number of k-gons in the graph will vary as the edge points change position, the total number of regions will stay constant as long as all internal vertices remain simple.

Scott R. Shannon, Image for n = 0.
Scott R. Shannon, Image for n = 1.
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.

Cf. A334698 (vertices), A367122 (edges), A255011, A367118.

Conjecture: a(n) = (17/2)*n^4 + 19*n^3 + (43/2)*n^2 + 14*n + 4.

a(n) = A367122(n) - A334698(n+1) + 1 by Euler's formula.

A367276 Place n equally spaced points on each side of a square, and join each of these points by a chord to the 3*n new points on the other three sides: sequence gives number of vertices in the resulting planar graph.

Original entry on oeis.org

4, 9, 69, 345, 1337, 2885, 7445, 12833, 23365, 36589, 64669, 80133, 138313, 176885, 233765, 312013, 455273, 513277, 741965, 819589, 1046245, 1310761, 1692961, 1772097, 2315289, 2713997, 3165125, 3552753, 4538845, 4602985, 6015561, 6432681, 7421345, 8550485, 9439621, 10063993, 12635769

Offset: 0

Scott R. Shannon, Nov 11 2023

nonn

We start with the four corner points of the square, and add n further points along each edge. Including the corner points, we end up with n+2 points along each edge, and the edge is divided into n+1 line segments.

Each of the n points added to an edge is joined by 3*n chords to the points that were added to the other three edges. There are 6*n^2 chords.

Scott R. Shannon, Image for n = 1.
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 10.

Cf. A367277 (interior vertices), A367278 (regions), A367279 (edges).
If the 4*n points are placed "in general position" instead of uniformly, we get sequences A334698, A367121, A367122.
If the 4*n points are placed uniformly and we also draw chords from the four corner points of the square to these 4*n points, we get A255011, A331448, A331449, A334690.

a(n) = A367279(n) - A367278(n) + 1 (Euler).

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.

Original entry on oeis.org

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).

A345649 Number of vertices formed when connecting all 4n points on the perimeter of an n X n square by infinite lines.

Original entry on oeis.org

0, 5, 61, 465, 1585, 5257, 9749, 24025, 38381, 67177, 100889, 176005, 210033, 360877, 450349, 589581, 779541, 1127509, 1251805, 1806061, 1970129, 2504401, 3116945, 4017701, 4163753, 5433657, 6335589

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Jun 21 2021

See A345459 for images of the polygons.

Cf. A345459 (number of polygons), A345650 (number of edges), A255011, A345648, A344993, A344857, A092098, A007678.

a(n) = A345650(n) - A345459(n) + 1.

A345650 Number of polygon edges formed when connecting all 4n points on the perimeter of an n X n square by infinite lines.

Original entry on oeis.org

0, 8, 140, 1032, 3608, 11308, 21892, 51000, 83404, 143900, 217728, 367132, 456008, 749328, 952236, 1251056, 1649708, 2327232, 2653900, 3717444, 4158448, 5243680, 6488208, 8241988, 8780976, 11235028, 13116156

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Jun 21 2021

See A345459 for images of the polygons.

Cf. A331448 (number inside the square), A345459 (number of polygons), A345649 (number of vertices), A255011, A345648, A344993, A344857, A092098, A007678.

a(n) = A345459(n) + A345649(n) - 1.

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.

cp's OEIS Frontend

A335353 Irregular table read by rows: Take a square and divide each of its sides into n equal parts giving a total of 4*n nodes, draw straight line segments from node k to node (k+n+1) mod 4*n, 0 <= k < 4*n. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

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A335354 a(n) is the number of edges in the central polygon formed in a square by dividing each of its sides into n equal parts giving a total of 4*n nodes and drawing straight line segments from node k to node (k+n+1) mod 4*n, 0 <= k < 4*n.

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A367121 Place n points in general position on each side of a square, and join every pair of the 4*n+4 boundary points by a chord; sequence gives number of regions in the resulting planar graph.

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A367276 Place n equally spaced points on each side of a square, and join each of these points by a chord to the 3*n new points on the other three sides: sequence gives number of vertices in the resulting planar graph.

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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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A345649 Number of vertices formed when connecting all 4n points on the perimeter of an n X n square by infinite lines.

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A345650 Number of polygon edges formed when connecting all 4n points on the perimeter of an n X n square by infinite lines.

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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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A335353 Irregular table read by rows: Take a square and divide each of its sides into n equal parts giving a total of 4n nodes, draw straight line segments from node k to node (k+n+1) mod 4n, 0 <= k < 4*n. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

A335354 a(n) is the number of edges in the central polygon formed in a square by dividing each of its sides into n equal parts giving a total of 4n nodes and drawing straight line segments from node k to node (k+n+1) mod 4n, 0 <= k < 4*n.