A331702 Number of distinct intersections among all circles that can be constructed on vertices of an n-sided regular polygon, using only a compass.

0, 2, 6, 40, 55, 145, 238, 584, 612, 1350, 1804, 2401, 3523, 5180, 6150, 9312, 11101, 13645, 17746, 22300, 25998, 33462, 39514, 43993, 55225, 66976, 74088, 88956, 102109, 111841, 133672, 155808, 170940, 198798, 220150, 243937, 275983, 313728, 338208, 382480, 419143, 448561, 507658

Offset: 1

Author

Matej Veselovac, Jan 25 2020

Keywords

• nonn

Comments

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)

Examples

			a(1)=0, we need at least two points to define a radius and a center.
a(2)=2, 2 circles constructed on segment endpoints intersect at 2 points.
a(3)=6, 3 circles on vertices of a triangle intersect at 6 distinct points.
a(4)=40, 8 circles can be constructed on vertices of a square and intersect at 40 distinct points.
a(5)=55, 10 circles can be constructed on vertices of a pentagon and intersect at 55 distinct points.
		

Links

• 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 = 5.
• Scott R. Shannon, Image for n = 6.
• Scott R. Shannon, Image for n = 7.
• Scott R. Shannon, Image for n = 8.
• Scott R. Shannon, Image for n = 9.
• Scott R. Shannon, Image for n = 10.
• Scott R. Shannon, Image for n = 11.
• Scott R. Shannon, Image for n = 12.
• Scott R. Shannon, Image for n = 18.
• Scott R. Shannon, Image for n = 25.
• N. J. A. Sloane, Illustration for A331702(4) = 40. Shows the planar graph. Annotated version of an illustration in the Math StackEchange link.
• Math StackExchange, Intersections of circles drawn on vertices of regular polygons, 2020.

Crossrefs

Cf. A093353, A359046 (regions), A359047 (edges), A359061 (k-gons), A358746.

Programs

GeoGebra

n = Slider(2, 10, 1);
C = Unique(RemoveUndefined(Flatten(Sequence(Sequence(Circle(Point({cos((2v Pi) / n), sin((2v Pi) / n)}), 2sin((c Pi) / n)), c, 1, floor(n / 2)), v, 1, n))));
I = Unique(RemoveUndefined(Flatten(Sequence(Sequence(Intersect(Element(C, i), Element(C, j)), j, 1, Length(C)), i, 1, Length(C)))));
a_n = Length(I);

Extensions

a(24)-a(30) from Giovanni Resta, Mar 27 2020

a(31)-a(43) from Scott R. Shannon, Dec 14 2022