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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).

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).

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

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

Conjectures: for odd values of n all vertices are simple, other than those defining the diameters of the circles. For n > 2 and (n-2) mod 4 = 0, T(n,2) = n. For n mod 4 = 0, T(n,2) = k*n, k>=2. For odd n, T(n,2) = 0.

See A358782 for more images of the k-gons.

The author thanks Zach Shannon some of whose code was used in the generation of this sequence.

The number of circles that cross to form the intersections follows a similar pattern to that seen in A371254; see that sequence for further information. The details of the crossing counts are given in A371377.

Sequence counts intersections among all distinct circles such that: A circle is defined by a pair of distinct points of a regular n-sided polygon. First point is the center of the circle, while the distance between the points defines the radius of the circle.

It seems one additional intersection exists at the center of the polygon if and only if n is a multiple of 6. From this and n symmetries of the n-sided regular polygon, it would follow that n divides either a(n) or a(n)-1, depending on whether n is a multiple of 6.

A093353(n-1) gives the number of unique circles whose intersections a(n) counts.

From Scott R. Shannon, Dec 15 2022 (Start)

The values for n which lead to all vertices, other than those defining the n-sided regular polygon, being simple start 2, 3, 4, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, ... . These are all prime values except for the prime squares 4 and 25 which also appear. It is likely all primes appear although what other values lead to only simple vertices is unknown. (End)

A circle is constructed for every pair of the n points, the first point defines the circle's center while the second the radius distance. The number of distinct circles constructed for n points is A001859(n-1).

No formula for a(n) is currently known.

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.

Other than for n = 3, 4, and 6, all graphs so far investigated in this sequence contain some internal vertices which are created from the intersections of both 2 and 3 arcs, i.e., no graph contains only simple intersections. This is in contrast to the case where the point pairs are connected by straight lines, see A007569 and A335102, where the odd-n graphs contain only simple intersections. See the attached images.

Other patterns for the intersection arc counts are also seen. If n is divisible by 3 then a central vertex is always present that is created from the crossing of n arcs. If n is divisible by 6, then internal vertices are present that are created from the crossing of 6 arcs. For n = 15 and n = 45, internal vertices are present that are created from the crossing of 5 arcs - it is likely all graphs with n = 15+30*k, k>=0, contain such vertices.

For n = 30, the graph also contains internal vertices that are created from the crossing of 9 arcs. It is likely that all graphs with n divisible by 30 contain such vertices. As the graphs created from the straight line diagonal intersections of the regular n-gon, see A007569, also have the maximum possible line intersection count of 7 when n is divisible by 30, it is plausible that 9 is the maximum possible arc intersection count for any internal vertex, other than the central vertex when n is divisible by 3.

Assuming these patterns hold for all n, is it possible that there is a general formula for the number of vertices, analogous to that in A007569 for the intersections of chords in a regular n-gon?