A359619 -id:A359619 - cp's OEIS frontend

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.

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.

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.

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.

A361623 Irregular table read by rows: T(n,k) is the number of k-gons, k>=2, 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.

Original entry on oeis.org

0, 40, 60, 12, 0, 484, 583, 160, 28, 8, 0, 2196, 2416, 804, 104, 28, 0, 5676, 6616, 2184, 460, 40, 8, 13456, 16936, 5236, 1340, 104, 12, 4, 27512, 35032, 11796, 2400, 320, 28, 0, 4, 0, 50688, 65044, 22536, 4632, 584, 60, 12, 4, 8, 84300, 105860, 38024, 8124, 1080, 108

Offset: 1

Scott R. Shannon, Mar 18 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).

See A354605 and A353782 for images of the vertices and regions.

			The table begins:
 0, 40, 60, 12;
 0, 484, 583, 160, 28, 8;
 0, 2196, 2416, 804, 104, 28;
 0, 5676, 6616, 2184, 460, 40;
 8, 13456, 16936, 5236, 1340, 104, 12;
 4, 27512, 35032, 11796, 2400, 320, 28, 0, 4;
 0, 50688, 65044, 22536, 4632, 584, 60, 12, 4;
 8, 84300, 105860, 38024, 8124, 1080, 108;
.
.

Cf. A354605 (vertices), A353782 (regions), A356358 (edges), A361622 (distinct circles), A359935, A359862, A359258, A359619, A359061.

Sum of row n = A353782(n).

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.

cp's OEIS Frontend

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.

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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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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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A361623 Irregular table read by rows: T(n,k) is the number of k-gons, k>=2, 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.

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

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