cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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.

Original entry on oeis.org

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

Views

Author

Scott R. Shannon, Nov 30 2022

Keywords

Comments

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

Links

Crossrefs

Cf. A358746 (vertices), A358783 (edges), A359009 (k-gons), A007678, A344857.
See allso A370976-A370979.

Formula

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

Views

Author

Scott R. Shannon, Nov 30 2022

Keywords

Comments

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

Links

Crossrefs

Cf. A358782 (regions), A358783 (edges), A359009 (k-gons), A007569, A146212.
See allso A370976-A370979.

Formula

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

Views

Author

Scott R. Shannon, Nov 30 2022

Keywords

Comments

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

Crossrefs

Cf. A358746 (vertices), A358782 (regions), A359009 (k-gons), A135565, A344899.
See allso A370976-A370979.

Formula

a(n) = A358746(n) + A358782(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

Views

Author

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

Keywords

Comments

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).

Links

Crossrefs

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

Formula

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

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

Views

Author

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

Keywords

Comments

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).

Links

Crossrefs

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

Formula

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

Views

Author

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

Keywords

Comments

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

Links

Crossrefs

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

Formula

a(n) = A358782(n) + n if n even, a(n) = A358782(n) + 3*n if n odd.
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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