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 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

%I

%S 0,2,6,40,55,145,238,584,612,1350,1804,2401,3523,5180,6150,9312,11101,

%T 13645,17746,22300,25998,33462,39514,43993,55225,66976,74088,88956,

%U 102109,111841

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

%C 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.

%C 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 or not n is a multiple of 6.

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

%H Math StackExchange, <a href="https://math.stackexchange.com/q/3518768">Intersections of circles drawn on vertices of regular polygons</a>, 2020.

%e a(1)=0, we need at least two points to define a radius and a center.

%e a(2)=2, 2 circles constructed on segment endpoints intersect at 2 points.

%e a(3)=6, 3 circles on vertices of a triangle intersect at 6 distinct points.

%e a(4)=40, 8 circles can be constructed on vertices of a square and intersect at 40 distinct points.

%e a(5)=55, 10 circles can be constructed on vertices of a pentagon and intersect at 55 distinct points.

%o (GeoGebra)

%o n = Slider(2, 10, 1);

%o 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))));

%o I = Unique(RemoveUndefined(Flatten(Sequence(Sequence(Intersect(Element(C, i), Element(C, j)), j, 1, Length(C)), i, 1, Length(C)))));

%o a_n = Length(I);

%Y Cf. A093353.

%K nonn,more

%O 1,2

%A _Matej Veselovac_, Jan 25 2020

%E a(24)-a(30) from _Giovanni Resta_, Mar 27 2020

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Last modified January 22 23:50 EST 2022. Contains 350504 sequences. (Running on oeis4.)