A358782 -id:A358782 - cp's OEIS frontend

A359033 Maximum number of sides in any region when the vertices of a regular n-gon are connected by circles and where the vertices lie at the ends of the circles' diameters (cf. A359009 and A358782).

2, 3, 3, 5, 12, 7, 6, 9, 20, 11, 6, 13, 28, 15, 8, 17, 36, 19, 8, 21, 44, 23, 10, 25, 52, 27, 8, 29, 60, 31, 10, 33, 68, 35, 10, 37, 76, 39, 10, 41, 84

Offset: 2

Scott R. Shannon, Dec 12 2022

See A358782 for more images of the n-gons.

Scott R. Shannon, Image for n = 18.
Scott R. Shannon, Image for n = 28.

Cf. A359009 (k-gons), A358782 (regions).

A358746 The number of vertices formed when every pair of n points, placed at the vertices of a regular n-gon, are connected by a circle and where the points lie at the ends of the circle's diameter.

Original entry on oeis.org

2, 6, 5, 55, 54, 252, 169, 747, 630, 1804, 1381, 3679, 3150, 6690, 5553, 11509, 9846, 18012, 15241, 27237, 24398, 39606, 33577, 56275, 50622, 77058, 69693, 102979, 94770, 135966, 124065, 175593, 162894, 222810, 205885, 279831, 260870, 347178, 321961, 424391, 399042

Offset: 2

Scott R. Shannon, Nov 30 2022

nonn

Conjecture: for odd values of n all vertices are simple, other than those defining the diameters of the circles. No formula for n, or only the odd values of n, is currently known.
The author thanks Zach Shannon some of whose code was used in the generation of this sequence.
If n is odd, the circle containing the initial n points is not part of the graph (compare A370976-A370979). - N. J. A. Sloane, Mar 25 2024

Scott R. Shannon, Image for n = 2. In this and other images the points defining the circle diameters are show as white dots.
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 = 6.
Scott R. Shannon, Image for n = 7.
Scott R. Shannon, Image for n = 8.
Scott R. Shannon, Image for n = 9.
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 16.
Scott R. Shannon, Image for n = 17.

Cf. A358782 (regions), A358783 (edges), A359009 (k-gons), A007569, A146212.

See allso A370976-A370979.

a(n) = A358783(n) - A358782(n) + 1 by Euler's formula.

A358783 The number of edges formed when every pair of n points, placed at the vertices of a regular n-gon, are connected by a circle and where the points lie at the ends of the circle's diameter.

Original entry on oeis.org

2, 12, 16, 120, 138, 532, 432, 1548, 1450, 3696, 3120, 7488, 6874, 13560, 12000, 23256, 20970, 36328, 32400, 54852, 51194, 79672, 70752, 113100, 105274, 154764, 144480, 206712, 195810, 272800, 255808, 352176, 335002, 446740, 422784, 560920, 534698, 695760, 659440, 850340, 815682

Offset: 2

Scott R. Shannon, Nov 30 2022

nonn

Conjecture: for odd values of n all vertices are simple, other than those defining the diameters of the circles. No formula for n, or only the odd values of n, is currently known.
See A358746 and A358782 for images of the circles.
The author thanks Zach Shannon some of whose code was used in the generation of this sequence.
If n is odd, the circle containing the initial n points is not part of the graph (compare A370976-A370979). - N. J. A. Sloane, Mar 25 2024

Cf. A358746 (vertices), A358782 (regions), A359009 (k-gons), A135565, A344899.

See allso A370976-A370979.

a(n) = A358746(n) + A358782(n) - 1 by Euler's formula.

A359009 Irregular table read by rows: T(n,k) is the number of k-gons formed, k>=2, when every pair of n points, placed at the vertices of a regular n-gon, are connected by a circle and where the points lie at the ends of the circle's diameter.

Original entry on oeis.org

1, 0, 7, 8, 4, 0, 40, 20, 6, 6, 72, 6, 0, 0, 0, 0, 0, 0, 0, 1, 0, 133, 98, 42, 7, 1, 16, 184, 56, 0, 8, 0, 342, 306, 99, 54, 0, 0, 1, 10, 510, 220, 60, 10, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 693, 858, 231, 88, 11, 11, 0, 0, 1, 24, 924, 612, 120, 60, 0, 1469, 1560, 455, 299, 13, 0, 0, 13, 0, 0, 1

Offset: 2

Scott R. Shannon, Dec 12 2022

Conjectures: for odd values of n all vertices are simple, other than those defining the diameters of the circles. For n > 2 and (n-2) mod 4 = 0, T(n,2) = n. For n mod 4 = 0, T(n,2) = k*n, k>=2. For odd n, T(n,2) = 0.
See A358782 for more images of the k-gons.
The author thanks Zach Shannon some of whose code was used in the generation of this sequence.

			The table begins:
1;
0, 7;
8, 4;
0, 40, 20, 6;
6, 72, 6, 0, 0, 0, 0, 0, 0, 0, 1;
0, 133, 98, 42, 7, 1;
16, 184, 56, 0, 8;
0, 342, 306, 99, 54, 0, 0, 1;
10, 510, 220, 60, 10, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
0, 693, 858, 231, 88, 11, 11, 0, 0, 1;
24, 924, 612, 120, 60;
0, 1469, 1560, 455, 299, 13, 0, 0, 13, 0, 0, 1;
14, 1806, 1428, 350, 98, 28, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \\
                0, 0, 0, 1;
0, 2550, 2910, 870, 405, 75, 60, 0, 0, 0, 0, 0, 0, 1;
32, 3280, 2000, 768, 352, 0, 16;
0, 4301, 4862, 1734, 680, 102, 34, 0, 17, 0, 17, 0, 0, 0, 0, 1;
18, 4878, 4482, 1332, 324, 54, 36, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \\
                0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
0, 6517, 7847, 2565, 1045, 190, 133, 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
80, 7340, 7040, 1920, 700, 0, 80;
0, 9723, 11487, 4515, 1491, 210, 168, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
.
.

Scott R. Shannon, Image for n = 13.
Scott R. Shannon, Image for n = 21.

Cf. A358782 (regions), A358746 (vertices), A358783 (edges), A331451, A344938.

Sum of row n = A358782(n).

A371374 Place n equally spaced points around the circumference of a circle and then, for each pair of points, draw two distinct circles, whose radii are the same as the first circle, such that both points lie on their circumferences. The sequence gives the total number of regions formed.

Original entry on oeis.org

1, 1, 9, 9, 51, 48, 211, 217, 612, 651, 1475, 1248, 3017, 3193, 5415, 5793, 9623, 9000, 15429, 15901, 23352, 24311, 34501, 33840, 49001, 50337, 67365, 69385, 91003, 87720, 120219, 123169, 155430, 159291, 198521, 198792, 250121, 256121, 310635, 317441, 382203, 382032, 465691, 473573

Offset: 1

Scott R. Shannon, Mar 20 2024

nonn

See A371373 and A371254 for further information. The details of the number of regions with k sides is given in A371376.

Scott R. Shannon, Image for n = 3. In this and other images the initial circle and n points are shown in white for clarity.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 6.
Scott R. Shannon, Image for n = 7.
Scott R. Shannon, Image for n = 8.
Scott R. Shannon, Image for n = 9.
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 11.
Scott R. Shannon, Image for n = 12.
Scott R. Shannon, Image for n = 15.
Scott R. Shannon, Image for n = 20.
Scott R. Shannon, Image for n = 24.
Scott R. Shannon, Image for n = 30.

Cf. A371373 (vertices), A371375 (edges), A371376 (k-gons), A371377 (vertex crossings), A371254, A371253, A006533, A358782, A359046.

a(n) = A371375(n) - A371373(n) + 1 by Euler's formula.

A359046 Number of distinct regions among all circles that can be constructed on vertices of an n-sided regular polygon, using only a compass.

Original entry on oeis.org

1, 3, 7, 45, 66, 186, 267, 657, 721, 1501, 1893, 2772, 3654, 5727, 6511, 9969, 11340, 14850, 18051, 23921, 26755, 35201, 39975, 47280, 55776, 69863, 75385, 93017, 102864, 117810, 134541, 161217, 172921, 205293, 221271, 252828, 277242, 322811, 341017, 393721, 420702, 466074, 509379

Offset: 1

Scott R. Shannon, Dec 14 2022

nonn

See A331702 for further details.

No formula for a(n) is currently known.

Scott R. Shannon, Image for n = 2. In this and other images the points defining the n-sided regular polygon are show as white dots.
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 = 6.
Scott R. Shannon, Image for n = 7.
Scott R. Shannon, Image for n = 8.
Scott R. Shannon, Image for n = 9.
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 11.
Scott R. Shannon, Image for n = 12.
Scott R. Shannon, Image for n = 18.
Scott R. Shannon, Image for n = 21.
Scott R. Shannon, Image for n = 25.

Cf. A331702 (vertices), A359047 (edges), A359061 (k-gons), A358782, A007678.

a(n) = A359047(n) - A331702(n) + 1 by Euler's formula.

A359061 Irregular table read by rows: T(n,k) is the number of k-gons formed, k>=2, among all circles that can be constructed on vertices of an n-sided regular polygon, using only a compass.

Original entry on oeis.org

3, 0, 7, 0, 16, 29, 0, 30, 35, 1, 0, 90, 96, 0, 105, 126, 35, 1, 0, 272, 304, 48, 32, 0, 1, 0, 315, 324, 81, 0, 0, 0, 1, 0, 460, 940, 60, 40, 0, 0, 0, 1, 0, 671, 858, 264, 88, 11, 0, 0, 0, 1, 0, 960, 1656, 108, 48, 0, 1144, 1807, 559, 130, 13, 0, 0, 0, 0, 0, 1, 0, 1960, 3136, 448, 168, 0, 14, 0, 0, 0, 0, 0, 1

Offset: 2

Scott R. Shannon, Dec 14 2022

See A331702 and A359046 for further details and images.

Conjecture: the only value for n which leads to the creation of 2-gons is n = 2. Despite values for n mod 6 = 0 forming intersecting arcs at the center of the n-gon, these are cut by other circles and thus create 3-gons or 4-gons. This is in contrast to values of n mod 4 = 0 in A359009 which do lead to the creation of 2-gons at the center of the figure from similar arcs.

			The table begins:
3;
0, 7;
0, 16, 29;
0, 30, 35, 1;
0, 90, 96;
0, 105, 126, 35, 1;
0, 272, 304, 48, 32, 0, 1;
0, 315, 324, 81, 0, 0, 0, 1;
0, 460, 940, 60, 40, 0, 0, 0, 1;
0, 671, 858, 264, 88, 11, 0, 0, 0, 1;
0, 960, 1656, 108, 48;
0, 1144, 1807, 559, 130, 13, 0, 0, 0, 0, 0, 1;
0, 1960, 3136, 448, 168, 0, 14, 0, 0, 0, 0, 0, 1;
0, 2100, 3270, 945, 180, 15, 0, 0, 0, 0, 0, 0, 0, 1;
0, 3088, 5584, 896, 368, 16, 16, 0, 0, 0, 0, 0, 0, 0, 1;
0, 3400, 5814, 1513, 493, 85, 34, 0, 0, 0, 0, 0, 0, 0, 0, 1;
0, 4536, 8712, 1224, 288, 54, 36;
0, 5586, 8797, 2774, 665, 76, 152, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
0, 7940, 12480, 2440, 960, 100, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
0, 7833, 14175, 3486, 1050, 147, 63, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
0, 10428, 19448, 3850, 1408, 22, 44, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

Scott R. Shannon, Image for n = 17.
Scott R. Shannon, Image for n = 23.
Scott R. Shannon, Image for n = 24.

Cf. A331702 (vertices), A359046 (regions), A359047 (edges), A359009, A358782, A007678.

Sum of row n = A359046(n).

A359253 Number of regions among all distinct circles that can be constructed from n equally spaced points along a line using only a compass.

Original entry on oeis.org

3, 14, 51, 116, 255, 466, 821, 1296, 2003, 2904, 4171, 5726, 7795, 10266, 13399, 17026, 21537, 26702, 32995, 40110, 48511, 57996, 69121, 81376, 95511, 111130, 128953, 148432, 170595

Offset: 2

Scott R. Shannon, Dec 22 2022

A circle is constructed for every pair of the n points, the first point defines the circle's center while the second the radius distance. The number of distinct circles constructed for n points is A001859(n-1).

No formula for a(n) is currently known.

Scott R. Shannon, Image for n = 2. In this and other images the n points are shown as white dots.
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 = 6.
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 11.
Scott R. Shannon, Image for n = 20.

Cf. A359252 (vertices), A359254 (edges), A359258 (k-gons), A001859, A290865, A359046, A358782.

a(n) = A359254(n) - A359252(n) + 1 by Euler's formula.

A359570 Number of regions after n iterations of constructing circles from all current vertices using only a compass, starting with one vertex. See the Comments.

Original entry on oeis.org

0, 1, 3, 21, 7169

Offset: 1

Scott R. Shannon, Jan 06 2023

See A359569 for further details and images.

Scott R. Shannon, Image for 3-region figure (black background). In this and other images the vertices used to construct the circles are shown as white dots.
Scott R. Shannon, Image for 3-region figure (white background).
Scott R. Shannon, Image for 21-region figure (black background).
Scott R. Shannon, Image for 21-region figure (white background).
Scott R. Shannon, Image for 7169-region figure (black background).
Scott R. Shannon, Image for 7169-region figure (white background).
N. J. A. Sloane, New Gilbreath Conjectures, Sum and Erase, Dissecting Polygons, and Other New Sequences, Doron Zeilberger's Exper. Math. Seminar, Rutgers, Sep 14 2023: Video, Slides, Updates. (Mentions this sequence.)

Cf. A359569 (vertices), A359571 (edges), A359619 (k-gons), A359253, A359046, A358782.

For n >= 3, a(n) = A359571(n) - A359569(n) + 1 by Euler's formula.

A370976 Let G_n denote the planar graph defined in A358746 with the addition, if n is odd, of the circle containing the initial n points; sequence gives the number of regions in G_n.

Original entry on oeis.org

1, 1, 10, 12, 71, 85, 288, 264, 811, 821, 1904, 1740, 3823, 3725, 6886, 6448, 11765, 11125, 18336, 17160, 27637, 26797, 40090, 37176, 56851, 54653, 77734, 74788, 103763, 101041, 136866, 131744, 176617, 172109, 223966, 216900, 281127, 273829, 348622, 337480, 425991, 416641

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 25 2024

nonn

If n is even the circle through the initial n points is already part of the graph.
In other words, draw a circle and place n equally spaced points around it; for each pair of poins X, Y, draw a circle with diameter XY; the union of these circles is the graph G_n.
For the numbers of vertices and edges in G_n see A358746 and A370977.
For other images for n even, see A358782 (for even n, A358782 and the present sequence agree).

Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 6.
Scott R. Shannon, Image for n = 7.
Scott R. Shannon, Image for n = 8.
Scott R. Shannon, Image for n = 11.

Cf. A358746, A358782, A358783, A370977, A370978, A370979.

a(n) = A358782(n) if n even, a(n) = A358782(n) + n if n odd.

cp's OEIS Frontend

A359033 Maximum number of sides in any region when the vertices of a regular n-gon are connected by circles and where the vertices lie at the ends of the circles' diameters (cf. A359009 and A358782).

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A358746 The number of vertices formed when every pair of n points, placed at the vertices of a regular n-gon, are connected by a circle and where the points lie at the ends of the circle's diameter.

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A358783 The number of edges formed when every pair of n points, placed at the vertices of a regular n-gon, are connected by a circle and where the points lie at the ends of the circle's diameter.

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A359009 Irregular table read by rows: T(n,k) is the number of k-gons formed, k>=2, when every pair of n points, placed at the vertices of a regular n-gon, are connected by a circle and where the points lie at the ends of the circle's diameter.

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A371374 Place n equally spaced points around the circumference of a circle and then, for each pair of points, draw two distinct circles, whose radii are the same as the first circle, such that both points lie on their circumferences. The sequence gives the total number of regions formed.

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A359046 Number of distinct regions among all circles that can be constructed on vertices of an n-sided regular polygon, using only a compass.

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A359061 Irregular table read by rows: T(n,k) is the number of k-gons formed, k>=2, among all circles that can be constructed on vertices of an n-sided regular polygon, using only a compass.

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A359253 Number of regions among all distinct circles that can be constructed from n equally spaced points along a line using only a compass.

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A359570 Number of regions after n iterations of constructing circles from all current vertices using only a compass, starting with one vertex. See the Comments.

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A370976 Let G_n denote the planar graph defined in A358746 with the addition, if n is odd, of the circle containing the initial n points; sequence gives the number of regions in G_n.

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