This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A359116 #42 Apr 22 2023 18:18:14 %S A359116 1,2,5,19,208,480,3011,7185,20169,35438,111232,162062,422841,633226, %T A359116 1024370,1576122,3315790,4240974,8204951,10654475,15310713 %N A359116 Mark the points of the Farey series F_n on a strip of paper and wrap it around a circle of circumference 1 so the endpoints 0 and 1 coincide; draw a chord between every pair of the Farey points; a(n) is the number of vertices in the resulting graph. %C A359116 Let F_n denote the Farey series of order n: F_1 = [0, 1]; F_2 = [0, 1/2, 1]; F_3 = [0, 1/3, 1/2, 2/3, 1], F_4 = [0, 1/4, 1/3, 1/2, 2/3, 3/4, 1], etc. In general F_n consists of the points i/j with 1 <= j <= n, 0 <= i <= j, gcd(i,j) = 1, sorted and duplicates removed. Alternatively, F_n = [A006842(n,k)/A006843(n,k), k = 1..A005728(n)]. %C A359116 The number of terms in F_n is A005728(n). Since the endpoints coincide when we wrap the series around the circle, there are M = A005728(n) - 1 vertices on the circumference. %C A359116 The planar graph we are studying, denoted by FR(n), is formed by drawing a chord between every pair of the M boundary points. FR stands for Farey Ring, a term suggested by the fairy rings found in nature. %C A359116 FR(n) is analogous to the planar graph formed by drawing chords between every pair of vertices of a regular n-gon, and studied in A006533 and A007678. The difference is that in FR(n) the vertices are not equally spaced around the circle. %C A359116 Just as in the case of the regular n-gon, when we count the regions in this graph, we may or may not include the regions that lie between the convex hull of the points and the bounding circle. %C A359116 The first non-simple vertices that do not lie on the y = 0 or x = 0 axes occur for n = 7. If we let A = (sin(3*Pi/14) + cos(Pi/7))/(cos(3*Pi/14) + sin(Pi/7)), and B = (cos(2*Pi/7)+1)/sin(2*Pi/7), then the x coordinate of these vertices is x = +-(A*cos(3*Pi/14) - sin(3*Pi/14) - 1)/(B + A), and their y coordinate is y = -B*x - 1. These values are approximately x = +-0.1930964297 and y = -0.5990311320. %H A359116 Tom Duff, <a href="/A359116/a359116.jpg">The Farey Ring graphs FR(2) to FR(10)</a> %H A359116 Tom Duff, <a href="/A359116/a359116_1.jpg">The Farey Ring graph FR(16)</a> %H A359116 Scott R. Shannon, <a href="/A359116/a359116.png">Image for n = 3</a>. %H A359116 Scott R. Shannon, <a href="/A359116/a359116_1.png">Image for n = 4</a>. %H A359116 Scott R. Shannon, <a href="/A359116/a359116_2.png">Image for n = 5</a>. %H A359116 Scott R. Shannon, <a href="/A359116/a359116_3.png">Image for n = 6</a>. %H A359116 Scott R. Shannon, <a href="/A359116/a359116_4.png">Image for n = 7</a>. The two non-simple vertices mentioned in the comments are the two yellow dots in the lower half of the figure on either side of the y axis. %H A359116 Scott R. Shannon, <a href="/A359116/a359116_5.png">Image for n = 8</a>. %H A359116 Scott R. Shannon, <a href="/A359116/a359116_6.png">Image for n = 9</a>. %H A359116 Scott R. Shannon, <a href="/A359116/a359116_7.png">Image for n = 10</a>. %F A359116 a(n) = A359118 - A359117 + 1 by Euler's formula. %Y A359116 Cf. A359117 (regions), A359118 (edges), A359119 (k-gons). %Y A359116 See also A358883, A005728, A006533, A006842, A006843, A007569, A007678. %K A359116 nonn,more %O A359116 1,2 %A A359116 _Scott R. Shannon_ and _N. J. A. Sloane_, Dec 16 2022