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.

1, 2, 5, 19, 208, 480, 3011, 7185, 20169, 35438, 111232, 162062, 422841, 633226, 1024370, 1576122, 3315790, 4240974, 8204951, 10654475, 15310713

Offset: 1

Author

Scott R. Shannon and N. J. A. Sloane, Dec 16 2022

Keywords

• nonn
• more

Comments

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)].

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.

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.

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.

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.

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.

Links

• Tom Duff, The Farey Ring graphs FR(2) to FR(10)
• Tom Duff, The Farey Ring graph FR(16)
• Scott R. Shannon, Image for n = 3.
• Scott R. Shannon, Image for n = 4.
• Scott R. Shannon, Image for n = 5.
• Scott R. Shannon, Image for n = 6.
• Scott R. Shannon, Image for n = 7. 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.
• Scott R. Shannon, Image for n = 8.
• Scott R. Shannon, Image for n = 9.
• Scott R. Shannon, Image for n = 10.

Crossrefs

Cf. A359117 (regions), A359118 (edges), A359119 (k-gons).

See also A358883, A005728, A006533, A006842, A006843, A007569, A007678.

Formula

a(n) = A359118 - A359117 + 1 by Euler's formula.