A359570 -id:A359570 - cp's OEIS frontend

A353782 Number of regions among all distinct circles that can be constructed from a point on the origin and n equally spaced points on each of the +x,-x,+y,-y coordinates axes using only a compass.

112, 1264, 5548, 14976, 37092, 77096, 143560, 237504

Offset: 1

Scott R. Shannon, Mar 13 2023

A circle is constructed for every pair of the 1 + 4n points, the first point defines the circle's center while the second the radius distance. The number of distinct circles constructed from the points is A361622(n).

No formula for a(n) is currently known.

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. A354605 (vertices), A356358 (edges), A361623 (k-gons), A361622 (distinct circles), A359933, A359860, A359253, A359570, A359046.

a(n) = A356358 - A354605(n) + 1 by Euler's formula.

A359569 Number of vertices 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

1, 2, 4, 14, 6562

Offset: 1

Scott R. Shannon, Jan 06 2023

Start with one vertex and a compass. Only a single circle can be drawn, using the vertex as its center, on whose circumference one additional vertex can be arbitrarily placed. From this single pair of vertices two circles can now be drawn, each circle's center being one vertex while the other defines its radius distance. These circles' intersections create two additional vertices, so now four vertices exist. Continue using all vertex pairs to draw distinct circles whose intersections create additional vertices for the next iteration. The sequence gives the number of vertices after n iterations of this process.

An estimate for a(6) can be obtained by calculating the number of distinct circles generated from the 6562 vertices of the fifth iteration - this is approximately 42.1 million circles. The 6562 vertices of the fifth iteration are created from 114 circles, implying the number of vertices per circle is about half the number of circles. Assuming this holds for the sixth iteration leads to an estimate for a(6) of about 886*10^12. The exact number is possibly within reach of numerical calculation, although obtaining a(7) would almost certainly require a theoretical approach.

Scott R. Shannon, Image for 4-vertex figure (black background).
Scott R. Shannon, Image for 4-vertex figure (white background)
Scott R. Shannon, Image for 14-vertex figure (black background).
Scott R. Shannon, Image for 14-vertex figure (white background)
Scott R. Shannon, Image for 6562-vertex figure (black background).
Scott R. Shannon, Image for 6562-vertex 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. A359570 (regions), A359571 (edges), A359619 (k-gons), A365669 (circles), A359252, A331702, A358746.

For n >= 3, a(n) = A359571(n) - A359570(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.

A365669 Number of distinct circles created after n iterations of constructing circles from all current vertices using only a compass, starting with one vertex.

Original entry on oeis.org

0, 1, 2, 6, 114, 42103152

Offset: 1

Scott R. Shannon, Sep 15 2023

See A359569 for further details and images.

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), A359570 (regions), A359571 (edges), A359619 (k-gons), A359931, A360350, A361622.

A359619 Irregular table read by rows: T(n,k) is the number of k-gons, k>=1, 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, 0, 0, 2, 1, 0, 1, 16, 4, 0, 16, 2470, 3599, 902, 168, 14

Offset: 1

Scott R. Shannon, Jan 07 2023

See A359569 and A359570 for further details and images.

			The table begins:
0;
1;
0, 0, 2, 1;
0, 1, 16, 4;
0, 16, 2470, 3599, 902, 168, 14;
.
.

Cf. A359569 (vertices), A359570 (regions), A359571 (edges), A359258, A359061, A359009.

Sum of row n = A359570(n);

A374337 Start with two vertices and draw a circle around each whose radius is the distance between the vertices. The sequence gives the number of regions constructed after n iterations of drawing circles with this same radius around every new vertex created from all circles' intersections.

Original entry on oeis.org

3, 11, 27, 55, 99, 145, 203, 277, 353, 441, 545, 651, 769, 903, 1039, 1187, 1351, 1517, 1695, 1889, 2085, 2293, 2517, 2743, 2981, 3235, 3491, 3759, 4043, 4329, 4627, 4941, 5257, 5585, 5929, 6275, 6633, 7007, 7383, 7771, 8175, 8581, 8999, 9433, 9869, 10317, 10781, 11247, 11725, 12219, 12715

Offset: 1

Scott R. Shannon, Jul 05 2024

nonn

See A374338 for further details.

Scott R. Shannon, Image for n = 1. In this and other images the initial vertices that form the circles' centers are shown as white dots.
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 = 16.

Cf. A374338 (vertices), A374339 (edges), A359570, A371374, A371253.

a(n) = A374339(n) - A374338(n) + 1, by Euler's formula.
Conjectured:
If n = 3*k + 1, k >= 0, a(n) = |(15*n^2 - 17*n - 7)/3|.
If n = 3*k, k >= 1, a(n) = (15*n^2 - 17*n - 3)/3.
If n = 3*k - 1, k >= 1, a(n) = (15*n^2 - 17*n + 7)/3.

A384702 On a 2 X n grid of vertices, draw a circle through every unordered triple of non-collinear vertices: a(n) is the number of distinct (finite) regions created.

Original entry on oeis.org

0, 1, 37, 245, 1205, 4213, 12261, 29742, 65507, 130824, 245325, 432262, 727259

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jun 07 2025

The infinite exterior region is not counted.

Scott R. Shannon, Image for a(3) = 37.
Scott R. Shannon, Image for a(4) = 245.
Scott R. Shannon, Image for a(5) = 1205.

Cf. A384700 (circles), A384701 (vertices), A384703 (edges), A359570, A374826, A374337, A372978.

a(n) = A384703(n) - A384701(n) + 1 by Euler's formula, for n > 1.

A385161 Place a point on the integer coordinates, up to |n|, along all four axial directions on a Cartesian plane, and then join a circle through every unordered triple of non-collinear points: a(n) is the number of distinct (finite) regions created.

Original entry on oeis.org

1, 117, 4713, 41173, 233365, 725081, 2323869

Offset: 1

Scott R. Shannon, Jul 10 2025

Scott R. Shannon, Image for n = 2. The 4 x 2 = 8 starting points are shown as white dots.
Scott R. Shannon, Image for n = 3. The 4 x 3 = 12 starting points are shown as white dots.

Cf. A385159 (total circles), A385160 (vertices), A385162 (edges), A384702, A374826, A372977, A359570.

a(n) = A385162(n) - A385160(n) + 1 by Euler's formula.

cp's OEIS Frontend

A353782 Number of regions among all distinct circles that can be constructed from a point on the origin and n equally spaced points on each of the +x,-x,+y,-y coordinates axes using only a compass.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A359569 Number of vertices after n iterations of constructing circles from all current vertices using only a compass, starting with one vertex. See the Comments.

Views

Author

Keywords

Comments

Links

Crossrefs

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.

Views

Author

Keywords

Comments

Crossrefs

Formula

A365669 Number of distinct circles created after n iterations of constructing circles from all current vertices using only a compass, starting with one vertex.

Views

Author

Keywords

Comments

Links

Crossrefs

A359619 Irregular table read by rows: T(n,k) is the number of k-gons, k>=1, after n iterations of constructing circles from all current vertices using only a compass, starting with one vertex. See the Comments.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A374337 Start with two vertices and draw a circle around each whose radius is the distance between the vertices. The sequence gives the number of regions constructed after n iterations of drawing circles with this same radius around every new vertex created from all circles' intersections.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A384702 On a 2 X n grid of vertices, draw a circle through every unordered triple of non-collinear vertices: a(n) is the number of distinct (finite) regions created.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A385161 Place a point on the integer coordinates, up to |n|, along all four axial directions on a Cartesian plane, and then join a circle through every unordered triple of non-collinear points: a(n) is the number of distinct (finite) regions created.

Views

Author

Keywords

Links

Crossrefs

Formula