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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).
25
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
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 A375817
KEYWORD
nonn,more
AUTHOR
STATUS
approved