A359569 -id:A359569 - cp's OEIS frontend

A354605 Number of vertices 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.

101, 1145, 5001, 13753, 34497, 72185, 135157, 224321

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.

Cf. A353782 (regions), A356358 (edges), A361623 (k-gons), A361622 (distinct circles), A359932, A359859, A359252, A359569, A331702.

a(n) = A356358 - A353782(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.

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.

A374338 Start with two vertices and draw a circle around each whose radius is the distance between the vertices. The sequence gives the number of vertices constructed after n iterations of drawing circles with this same radius around every new vertex created from all circles' intersections. See the Comments.

Original entry on oeis.org

4, 8, 14, 24, 34, 46, 62, 78, 96, 118, 140, 164, 192, 220, 250, 284, 318, 354, 394, 434, 476, 522, 568, 616, 668, 720, 774, 832, 890, 950, 1014, 1078, 1144, 1214, 1284, 1356, 1432, 1508, 1586, 1668, 1750, 1834, 1922, 2010, 2100, 2194, 2288, 2384, 2484, 2584, 2686, 2792

Offset: 1

Scott R. Shannon, Jul 05 2024

nonn

Start with two vertices and, using each as the center, draw a circle around each whose radius is the distance between the vertices. These circles' intersections create two additional vertices, so after the first iteration four vertices exist. Using these four vertices as centers draw four new circles whose radius is the same as the distance between the initial two vertices. These circles' intersections create eight new vertices. Repeat this process n times; the sequence gives the number of vertices after n iterations.

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

Cf. A374337 (regions), A374339 (edges), A359569, A371373, A371254.

a(n) = A374339(n) - A374337(n) + 1, by Euler's formula.
Conjectured:
If n = 3*k + 1, k >= 0, a(n) = (3*n^2 + 5*n + 4)/3.
If n = 3*k, k >= 1, a(n) = (3*n^2 + 5*n)/3.
If n = 3*k - 1, k >= 1, a(n) = (3*n^2 + 5*n + 2)/3.

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);

A384701 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 points where circles intersect.

Original entry on oeis.org

2, 4, 18, 172, 978, 3672, 11034, 27241, 60804, 122741, 232138, 412263, 697058

Offset: 1

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

Scott R. Shannon, Image for a(2) = 4.
Scott R. Shannon, Image for a(3) = 18.
Scott R. Shannon, Image for a(4) = 172.
Scott R. Shannon, Image for a(5) = 978.

Cf. A384700 (circles), A384702 (regions), A384703 (edges), A359569, A374825, A374338, A373106.

a(n) = A384703(n) - A384702(n) + 1 by Euler's formula, for n > 1.

A385160 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 points where circles intersect.

Original entry on oeis.org

4, 68, 4244, 38100, 222300, 695544, 2252764

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), A385161 (regions), A385162 (edges), A384703, A383461, A374825, A359569.

a(n) = A385162(n) - A385161(n) + 1 by Euler's formula.

cp's OEIS Frontend

A354605 Number of vertices 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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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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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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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.

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A384701 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 points where circles intersect.

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Formula

A385160 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 points where circles intersect.

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Author

Keywords

Links

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