cp's OEIS Frontend

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).

%I A358886 #20 Dec 06 2022 19:33:37
%S A358886 4,56,1040,6064,53104,115496,629920,1457744,3952264,6835568
%N 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).
%C A358886 The number of points along each edge is given by A005728(n).
%C A358886 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.
%H A358886 Scott R. Shannon, <a href="/A358886/a358886.jpg">Image for n = 1</a>.
%H A358886 Scott R. Shannon, <a href="/A358886/a358886_1.jpg">Image for n = 2</a>.
%H A358886 Scott R. Shannon, <a href="/A358886/a358886_2.jpg">Image for n = 3</a>.
%H A358886 Scott R. Shannon, <a href="/A358886/a358886_3.jpg">Image for n = 4</a>.
%H A358886 Scott R. Shannon, <a href="/A358886/a358886_4.jpg">Image for n = 5</a>.
%H A358886 Scott R. Shannon, <a href="/A358886/a358886_5.jpg">Image for n = 6</a>.
%H A358886 Wikipedia, <a href="https://en.wikipedia.org/wiki/Farey_sequence">Farey sequence</a>.
%F A358886 a(n) = A358888(n) - A358887(n) + 1 by Euler's formula.
%Y A358886 Cf. A358888 (edges), A358887 (vertices), A358889 (k-gons), A006842, A006843, A005728, A358882.
%Y A358886 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.
%K A358886 nonn,more
%O A358886 1,1
%A A358886 _Scott R. Shannon_ and _N. J. A. Sloane_, Dec 05 2022