A365669 -id:A365669 - 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.

A372682 Number of distinct circles that can be constructed from the 3 vertices and the equally spaced 3*n points placed on the sides of an equilateral triangle, using only a compass.

Original entry on oeis.org

3, 15, 36, 69, 123, 180, 264, 339, 453, 549, 702, 807, 999, 1128, 1329, 1494, 1749, 1935, 2214, 2373, 2682, 2940, 3288, 3483

Offset: 0

Scott R. Shannon, May 10 2024

See A372614 for images of the circles.

Cf. A372614, A372615, A359931, A360350, A361622, A365669.

A372735 Number of distinct circles that can be constructed from the 3 vertices and the equally spaced 3n points placed on the sides of an equilateral triangle when every pair of the 3 + 3n points are connected by a circle and where the points lie at the ends of the circle's diameter.

Original entry on oeis.org

3, 15, 34, 63, 99, 148, 201, 267, 340, 423, 513, 616, 723, 843, 970, 1107, 1251, 1408, 1569, 1743, 1924, 2115, 2313, 2524, 2739, 2967, 3202, 3447, 3699

Offset: 1

Scott R. Shannon, May 11 2024

See A372731 for images of the circles.

Cf. A372731, A372732, A372682, A372614, A372615, A359931, A360350, A361622, A365669.

A372981 Number of distinct circles that can be constructed from the 4 vertices and the equally spaced 4*n points placed on the sides of a square, using only a compass.

Original entry on oeis.org

8, 32, 88, 160, 264, 400, 576, 732, 968, 1184, 1480, 1728, 2104, 2424, 2840, 3196, 3688, 4088, 4640, 5048, 5704, 6248, 6904, 7364

Offset: 0

Scott R. Shannon, May 19 2024

A circle is constructed for every pair of the 4 + 4*n points, the first point defines the circle's center while the second the radius distance.

See A372978 for images of the circles.

Cf. A372978, A372977, A372979, A359931, A372682, A361622, A365669.

A373110 Number of 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

5, 22, 54, 99, 159, 232, 320, 421, 537, 666, 810, 967, 1139, 1324, 1524, 1737, 1965, 2206, 2462, 2731, 3015, 3312, 3624, 3949

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.

See A373106 and A373107 for images of the circles.

Cf. A373106, A373107, A373108, A372981, A372735, A359931, A360350, A361622, A365669.

Conjectured:
For even n, a(n) = (14*n^2 + 21*n + 10)/2.
For odd n, a(n) = (14*n^2 + 21*n + 9)/2.

A384700 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 circles created.

Original entry on oeis.org

0, 1, 9, 24, 52, 93, 153, 232, 336, 465, 625, 816, 1044, 1309, 1617, 1968, 2368, 2817, 3321, 3880, 4500, 5181, 5929

Offset: 1

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

Scott R. Shannon, Image for a(3) = 9.

Cf. A384701 (vertices), A384702 (regions), A384703 (edges), A365669, A374338, A373110, A372981,

Conjecture:
for even n, a(n) = n^3/2 - n^2/4 - n,
for odd n > 1, a(n) = n^3/2 - n^2/4 - n + 3/4.

A385159 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 circles created.

Original entry on oeis.org

1, 18, 99, 280, 633, 1098, 1915, 2928, 4329, 6010, 8331, 10752, 14113, 17778, 21987

Offset: 1

Scott R. Shannon, Jun 20 2025

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

Cf. A385160 (vertices), A385161 (regions), A385162 (edges), A361622, A384700, A373110, A372735, A365669.

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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A372682 Number of distinct circles that can be constructed from the 3 vertices and the equally spaced 3*n points placed on the sides of an equilateral triangle, using only a compass.

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A372735 Number of distinct circles that can be constructed from the 3 vertices and the equally spaced 3*n points placed on the sides of an equilateral triangle when every pair of the 3 + 3*n points are connected by a circle and where the points lie at the ends of the circle's diameter.

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A372981 Number of distinct circles that can be constructed from the 4 vertices and the equally spaced 4*n points placed on the sides of a square, using only a compass.

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Comments

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A373110 Number of 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

Crossrefs

Formula

A384700 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 circles created.

Views

Author

Keywords

Links

Crossrefs

Formula

A385159 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 circles created.

Views

Author

Keywords

Links

Crossrefs

A372735 Number of distinct circles that can be constructed from the 3 vertices and the equally spaced 3n points placed on the sides of an equilateral triangle when every pair of the 3 + 3n points are connected by a circle and where the points lie at the ends of the circle's diameter.

A373110 Number of 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.