A358783 -id:A358783 - cp's OEIS frontend

A358782 The number of regions 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.

1, 7, 12, 66, 85, 281, 264, 802, 821, 1893, 1740, 3810, 3725, 6871, 6448, 11748, 11125, 18317, 17160, 27616, 26797, 40067, 37176, 56826, 54653, 77707, 74788, 103734, 101041, 136835, 131744, 176584, 172109, 223931, 216900, 281090, 273829, 348583, 337480, 425950, 416641

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 = 11.
Scott R. Shannon, Image for n = 12.
Scott R. Shannon, Image for n = 17.
Scott R. Shannon, Image for n = 20.
Scott R. Shannon, Image for n = 23.

Cf. A358746 (vertices), A358783 (edges), A359009 (k-gons), A007678, A344857.

See allso A370976-A370979.

a(n) = A358783(n) - A358746(n) + 1 by Euler's formula.

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.

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).

A371375 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 (curved) edges formed.

Original entry on oeis.org

1, 2, 12, 12, 75, 66, 350, 360, 1071, 1150, 2684, 2148, 5603, 5950, 10110, 10928, 18309, 16830, 29564, 30500, 44961, 46882, 66746, 64872, 95125, 97786, 131112, 135156, 177567, 169770, 235042, 240928, 304359, 312086, 389340, 388764, 491175, 503158, 610662, 624280, 752145, 749742, 917276

Offset: 1

Scott R. Shannon, Mar 20 2024

nonn

See A371373 and A371374 for images of the graphs.

Cf. A371373 (vertices), A371374 (regions), A371376 (k-gons), A371377 (vertex crossings), A371255, A135565, A358783, A359047.

a(n) = A371373(n) + A371374(n) - 1 by Euler's formula.

A359047 Number of distinct edges 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, 4, 12, 84, 120, 330, 504, 1240, 1332, 2850, 3696, 5172, 7176, 10906, 12660, 19280, 22440, 28494, 35796, 46220, 52752, 68662, 79488, 91272, 111000, 136838, 149472, 181972, 204972, 229650, 268212, 317024, 343860, 404090, 441420, 496764, 553224, 636538, 679224, 776200, 839844, 914634, 1017036

Offset: 1

Scott R. Shannon, Dec 14 2022

nonn

See A331702 and A359046 for further details and images.

No formula for a(n) is currently known.

Cf. A331702 (vertices), A359046 (regions), A359061 (k-gons), A358783, A135565.

a(n) = A331702(n) + A359046(n) - 1 by Euler's formula.

A359254 Number of edges 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

4, 26, 96, 216, 480, 882, 1564, 2464, 3804, 5502, 7912, 10864, 14816, 19526, 25508, 32392, 40992, 50818, 62852, 76432, 92460, 110590, 131904, 155306, 182316, 212188, 246316, 283586, 326100

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).

See A359252 and A359253 for images of the circles.

Cf. A359252 (vertices), A359253 (regions), A359258 (k-gons), A001859, A290866, A359047, A358783.

a(n) = A359252(n) + A359253(n) - 1 by Euler's formula.

A359571 Number of (curved) edges 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, 6, 34, 13730

Offset: 1

Scott R. Shannon, Jan 06 2023

See A359569 and A359570 for further details and images.

Cf. A359569 (vertices), A359570 (regions), A359619 (k-gons), A359254, A359047, A358783.

For n >= 3, a(n) = A359569(n) + A359570(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.

A370979 Draw a regular n-gon and the enclosing circle, then for each pair of vertices X, Y, draw a circle with diameter XY; the union of these figures is the graph H_n; sequence gives number of edges in H_n.

Original entry on oeis.org

1, 4, 21, 20, 135, 144, 553, 440, 1575, 1460, 3729, 3132, 7527, 6888, 13605, 12016, 23307, 20988, 36385, 32420, 54915, 51216, 79741, 70776, 113175, 105300, 154845, 144508, 206799, 195840, 272893, 255840, 352275, 335036, 446845, 422820, 561031, 534736, 695877, 659480, 850463, 815724

Offset: 1

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

nonn

For the numbers of vertices and regions in G_n see A358746 and A370978.

H_n is the union of the graph G_n defined in A370976 and the polygon through the initial n points.

Cf. A358746, A358782, A358783, A370976, A370977, A370978.

a(n) = A358783(n) if n even, a(n) = A358783(n) + n if n odd.

a(n) = A358783(n) + n if n even, a(n) = A358783(n) + 3*n if n odd.

A373108 Number of edges among all distinct circles that can be constructed from the 4 vertices and the equally spaced 4n points placed on the sides of a square when every pair of the 4 + 4n points are connected by a circle and where the points lie at the ends of the circle's diameter.

Original entry on oeis.org

16, 196, 1608, 5784, 16848, 37300, 78420, 136920, 233336, 363200, 565700

Offset: 0

Scott R. Shannon, May 25 2024

A circle is constructed for every pair of the 4 + 4*n points, the two points lying at the ends of a diameter of the circle.

See A373106 and A373107 for images of the circles.

Cf. A373106 (vertices), A373107 (regions), A373109 (k-gons), A373110 (circles), A372979, A372733, A358783, A362235, A360353.

a(n) = A373106(n) + A373107(n) - 1 by Euler's formula.

cp's OEIS Frontend

A358782 The number of regions 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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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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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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A371375 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 (curved) edges formed.

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Author

Keywords

Comments

Crossrefs

Formula

A359047 Number of distinct edges among all circles that can be constructed on vertices of an n-sided regular polygon, using only a compass.

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A359254 Number of edges among all distinct circles that can be constructed from n equally spaced points along a line using only a compass.

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A359571 Number of (curved) edges 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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Formula

A370979 Draw a regular n-gon and the enclosing circle, then for each pair of vertices X, Y, draw a circle with diameter XY; the union of these figures is the graph H_n; sequence gives number of edges in H_n.

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A373108 Number of edges among all distinct circles that can be constructed from the 4 vertices and the equally spaced 4*n points placed on the sides of a square when every pair of the 4 + 4*n points are connected by a circle and where the points lie at the ends of the circle's diameter.

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A373108 Number of edges among all distinct circles that can be constructed from the 4 vertices and the equally spaced 4n points placed on the sides of a square when every pair of the 4 + 4n points are connected by a circle and where the points lie at the ends of the circle's diameter.