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%I #53 Dec 15 2022 05:10:23
%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 102109,111841,133672,155808,170940,198798,220150,243937,275983,313728,338208,382480,419143,448561,507658
%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 n is a multiple of 6.
%C A093353(n-1) gives the number of unique circles whose intersections a(n) counts.
%C From _Scott R. Shannon_, Dec 15 2022 (Start)
%C 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)
%H Scott R. Shannon, <a href="/A331702/a331702.jpg">Image for n = 2</a>.
%H Scott R. Shannon, <a href="/A331702/a331702_1.jpg">Image for n = 3</a>.
%H Scott R. Shannon, <a href="/A331702/a331702_2.jpg">Image for n = 4</a>.
%H Scott R. Shannon, <a href="/A331702/a331702_3.jpg">Image for n = 5</a>.
%H Scott R. Shannon, <a href="/A331702/a331702_4.jpg">Image for n = 6</a>.
%H Scott R. Shannon, <a href="/A331702/a331702_5.jpg">Image for n = 7</a>.
%H Scott R. Shannon, <a href="/A331702/a331702_6.jpg">Image for n = 8</a>.
%H Scott R. Shannon, <a href="/A331702/a331702_7.jpg">Image for n = 9</a>.
%H Scott R. Shannon, <a href="/A331702/a331702_8.jpg">Image for n = 10</a>.
%H Scott R. Shannon, <a href="/A331702/a331702_9.jpg">Image for n = 11</a>.
%H Scott R. Shannon, <a href="/A331702/a331702_10.jpg">Image for n = 12</a>.
%H Scott R. Shannon, <a href="/A331702/a331702_11.jpg">Image for n = 18</a>.
%H Scott R. Shannon, <a href="/A331702/a331702_12.jpg">Image for n = 25</a>.
%H N. J. A. Sloane, <a href="/A331702/a331702_2.pdf">Illustration for A331702(4) = 40</a>. Shows the planar graph. Annotated version of an illustration in the Math StackEchange link.
%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, A359046 (regions), A359047 (edges), A359061 (k-gons), A358746.
%K nonn
%O 1,2
%A _Matej Veselovac_, Jan 25 2020
%E a(24)-a(30) from _Giovanni Resta_, Mar 27 2020
%E a(31)-a(43) from _Scott R. Shannon_, Dec 14 2022
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