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

1,1

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

Table of n, a(n) for n=1..10.

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.

Formula

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

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.

Sequence in context: A078533 A299302 A327199 * A371678 A009058 A276363

Adjacent sequences: A358883 A358884 A358885 * A358887 A358888 A358889

Keyword

nonn,more

Author

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

Status

approved