

A331702


Number of distinct intersections among all circles that can be constructed on vertices of a nsided regular polygon, using only a compass.


0



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
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OFFSET

1,2


COMMENTS

Sequence counts intersections among all distinct circles such that: A circle is defined by a pair of distinct points of a regular nsided 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 nsided regular polygon, it would follow that n divides either a(n) or a(n)1, depending on whether or not n is a multiple of 6.
A093353(n1) gives the number of unique circles whose intersections a(n) counts.


LINKS

Table of n, a(n) for n=1..30.
Math StackExchange, Intersections of circles drawn on vertices of regular polygons, 2020.


EXAMPLE

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.


PROG

(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);


CROSSREFS

Cf. A093353.
Sequence in context: A132192 A340299 A068207 * A288491 A212883 A336959
Adjacent sequences: A331699 A331700 A331701 * A331703 A331704 A331705


KEYWORD

nonn,more


AUTHOR

Matej Veselovac, Jan 25 2020


EXTENSIONS

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


STATUS

approved



