A371254 - OEIS

%I #33 Mar 25 2024 09:59:27

%S 1,2,4,4,15,7,70,64,208,220,550,397,1131,1162,1981,2128,3723,3259,

%T 5966,6000,9010,9240,13524,12745,19325,19266,26434,26684,35931,33301,

%U 47368,47616,61216,61676,78330,76789,98901,99674,122656,123560

%N Number of vertices formed when n equally spaced points are placed around a circle and all pairs of points are joined by an interior arc whose radius equals the circle's radius.

%C Other than for n = 3, 4, and 6, all graphs so far investigated in this sequence contain some internal vertices which are created from the intersections of both 2 and 3 arcs, i.e., no graph contains only simple intersections. This is in contrast to the case where the point pairs are connected by straight lines, see A007569 and A335102, where the odd-n graphs contain only simple intersections. See the attached images.

%C Other patterns for the intersection arc counts are also seen. If n is divisible by 3 then a central vertex is always present that is created from the crossing of n arcs. If n is divisible by 6, then internal vertices are present that are created from the crossing of 6 arcs. For n = 15 and n = 45, internal vertices are present that are created from the crossing of 5 arcs - it is likely all graphs with n = 15+30*k, k>=0, contain such vertices.

%C For n = 30, the graph also contains internal vertices that are created from the crossing of 9 arcs. It is likely that all graphs with n divisible by 30 contain such vertices. As the graphs created from the straight line diagonal intersections of the regular n-gon, see A007569, also have the maximum possible line intersection count of 7 when n is divisible by 30, it is plausible that 9 is the maximum possible arc intersection count for any internal vertex, other than the central vertex when n is divisible by 3.

%C Assuming these patterns hold for all n, is it possible that there is a general formula for the number of vertices, analogous to that in A007569 for the intersections of chords in a regular n-gon?

%H B. Poonen and M. Rubinstein, <a href="https://arxiv.org/abs/math/9508209">The Number of Intersection Points Made by the Diagonals of a Regular Polygon</a>, arXiv:math/9508209v3 [math.MG], 1995-2006.

%H Scott R. Shannon, <a href="/A371254/a371254.jpg">Image for n = 3</a>.

%H Scott R. Shannon, <a href="/A371254/a371254_1.jpg">Image for n = 4</a>.

%H Scott R. Shannon, <a href="/A371254/a371254_2.jpg">Image for n = 5</a>.

%H Scott R. Shannon, <a href="/A371254/a371254_3.jpg">Image for n = 6</a>.

%H Scott R. Shannon, <a href="/A371254/a371254_4.jpg">Image for n = 7</a>.

%H Scott R. Shannon, <a href="/A371254/a371254_5.jpg">Image for n = 8</a>.

%H Scott R. Shannon, <a href="/A371254/a371254_6.jpg">Image for n = 9</a>.

%H Scott R. Shannon, <a href="/A371254/a371254_7.jpg">Image for n = 10</a>.

%H Scott R. Shannon, <a href="/A371254/a371254_8.jpg">Image for n = 11</a>.

%H Scott R. Shannon, <a href="/A371254/a371254_9.jpg">Image for n = 12</a>.

%H Scott R. Shannon, <a href="/A371254/a371254_10.jpg">Image for n = 15</a>. Note the 5 arc intersections shown in green.

%H Scott R. Shannon, <a href="/A371254/a371254_11.jpg">Image for n = 20</a>.

%H Scott R. Shannon, <a href="/A371254/a371254_12.jpg">Image for n = 24</a>.

%H Scott R. Shannon, <a href="/A371254/a371254_14.jpg">Image for n = 30</a>. Note the 9 arc intersections shown in violet.

%F a(n) = A371255(n) - A371253(n) + 1 by Euler's formula.

%Y Cf. A371253 (regions), A371255 (edges), A371274 (k-gons), A371264 (vertex crossings), A370980 (number of circles), A371373 (complete circles), A007569, A335102, A358746, A331702, A359252.

%K nonn

%O 1,2

%A _Scott R. Shannon_, Mar 16 2024