login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A331702 Number of distinct intersections among all circles that can be constructed on vertices of a n-sided 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 (list; graph; refs; listen; history; text; internal format)
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 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 or not n is a multiple of 6.

A093353(n-1) 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 4 08:36 EST 2021. Contains 349480 sequences. (Running on oeis4.)