A358886 Number of regions formed inside a square with edge length 1 by the straight line segments mutually connecting all vertices and points that divide the sides into segments with lengths equal to the Farey series of order n = A006842(n,k)/A006843(n,k), k = 1..A005728(n).

4, 56, 1040, 6064, 53104, 115496, 629920, 1457744, 3952264, 6835568

Offset: 1

Author

Scott R. Shannon and N. J. A. Sloane, Dec 05 2022

Keywords

• nonn
• more

Comments

The number of points along each edge is given by A005728(n).

We call this graph the Completed Farey Diagram of order (n,k). The (ordinary) Farey diagram Farey(n,k) is a subgraph. In the latter graph, not all pairs of boundary points are joined by lines.

Links

• Scott R. Shannon, Image for n = 1.
• Scott R. Shannon, Image for n = 2.
• 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.
• Wikipedia, Farey sequence.

Crossrefs

Cf. A358888 (edges), A358887 (vertices), A358889 (k-gons), A006842, A006843, A005728, A358882.

The Farey Diagrams Farey(m,n) are studied in A358298-A358307 and A358882-A358885, the Completed Farey Diagrams of order (m,n) in A358886-A358889.

Formula

a(n) = A358888(n) - A358887(n) + 1 by Euler's formula.