A358782 -id:A358782 - cp's OEIS frontend

A371253 Number of regions formed when n equally spaced points are placed around a circle and all pairs of points are joined by an interior arc whose radius equals the circle's radius.

1, 1, 6, 5, 26, 18, 99, 89, 270, 271, 650, 516, 1288, 1303, 2250, 2337, 4047, 3636, 6404, 6401, 9597, 9769, 14261, 13632, 20251, 20125, 27594, 27749, 37324, 35040, 49043, 49185, 63228, 63547, 80676, 79380, 101640, 102259, 125853, 126561

Offset: 1

Scott R. Shannon, Mar 16 2024

nonn

See A371254 for further information.

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 = 16.
Scott R. Shannon, Image for n = 20.
Scott R. Shannon, Image for n = 24.

Cf. A371254 (vertices), A371255 (edges), A371274 (k-gons), A370980 (number of circles), A371374 (complete circles), A006533, A358782, A359046, A359253, A007678.

a(n) = A371255(n) - A371254(n) + 1 by Euler's 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.

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.

A373107 Number of regions 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

12, 136, 932, 3252, 9144, 20032, 41260, 71832, 121460, 188656, 291488

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.

Scott R. Shannon, Image for n = 0. In this and other images the 4 + 4*n vertices forming the square are drawn as white dots for clarity.
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 = 7.

Cf. A373106 (vertices), A373108 (edges), A373109 (k-gons), A373110 (circles), A372978, A372732, A358782, A362234, A360352.

a(n) = A373108(n) - A373106(n) + 1 by Euler's formula.

A370977 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 edges in G_n.

Original entry on oeis.org

1, 2, 15, 16, 125, 138, 539, 432, 1557, 1450, 3707, 3120, 7501, 6874, 13575, 12000, 23273, 20970, 36347, 32400, 54873, 51194, 79695, 70752, 113125, 105274, 154791, 144480, 206741, 195810, 272831, 255808, 352209, 335002, 446775, 422784, 560957, 534698, 695799, 659440, 850381, 815682

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 regions in G_n see A358746 and A370976.
For other images for n even, see A358746 (for even n, A358783 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 = 5.
Scott R. Shannon, Image for n = 6.
Scott R. Shannon, Image for n = 7.
Scott R. Shannon, Image for n = 8.

Cf. A358746, A358782, A358783, A370976, A370978, A370979.

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

A370978 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 regions in H_n.

Original entry on oeis.org

1, 3, 16, 16, 81, 91, 302, 272, 829, 831, 1926, 1752, 3849, 3739, 6916, 6464, 11799, 11143, 18374, 17180, 27679, 26819, 40136, 37200, 56901, 54679, 77788, 74816, 103821, 101071, 136928, 131776, 176683, 172143, 224036, 216936, 281201, 273867, 348700, 337520, 426073, 416683

Offset: 1

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

nonn

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

Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 5.

Cf. A358746, A358782, A358783, A370976, A370977, A370979.

a(n) = A358782(n) + n if n even, a(n) = A358782(n) + 3*n if n odd.

cp's OEIS Frontend

A371253 Number of regions formed when n equally spaced points are placed around a circle and all pairs of points are joined by an interior arc whose radius equals the circle's radius.

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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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Formula

A373107 Number of regions 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.

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Author

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Formula

A370977 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 edges in G_n.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A370978 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 regions in H_n.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

cp's OEIS Frontend

A371253 Number of regions formed when n equally spaced points are placed around a circle and all pairs of points are joined by an interior arc whose radius equals the circle's radius.

Views

Author

Keywords

Comments

Links

Crossrefs

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.

Views

Author

Keywords

Comments

Crossrefs

Formula

A373107 Number of regions 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.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A370977 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 edges in G_n.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A370978 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 regions in H_n.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A373107 Number of regions 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.