%I #8 Sep 27 2023 14:56:52
%S 3,10,148,1111,9568,23770,126187,308401,855145,1521733,4591405,6831040
%N Number of vertices formed inside a triangle 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 The number of points along each edge is given by A005728(n).
%H Scott R. Shannon, <a href="/A358949/a358949.png">Image for n = 2</a>.
%H Scott R. Shannon, <a href="/A358949/a358949_1.png">Image for n = 3</a>.
%H Scott R. Shannon, <a href="/A358949/a358949_2.png">Image for n = 4</a>.
%H Scott R. Shannon, <a href="/A358949/a358949_3.png">Image for n = 5</a>.
%H Scott R. Shannon, <a href="/A358949/a358949_4.png">Image for n = 6</a>.
%H N. J. A. Sloane, New Gilbreath Conjectures, Sum and Erase, Dissecting Polygons, and Other New Sequences, Doron Zeilberger's Exper. Math. Seminar, Rutgers, Sep 14 2023: <a href="https://vimeo.com/866583736?share=copy">Video</a>, <a href="http://neilsloane.com/doc/EMSep2023.pdf">Slides</a>, <a href="http://neilsloane.com/doc/EMSep2023.Updates.txt">Updates</a>. (Mentions this sequence.)
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Farey_sequence">Farey sequence</a>.
%F a(n) = A358950(n) - A358948(n) + 1 by Euler's formula.
%Y Cf. A358948 (regions), A358950 (edges), A358951 (k-gons), A358887, A006842, A006843, A005728, A358882.
%K nonn,more
%O 1,1
%A _Scott R. Shannon_ and _N. J. A. Sloane_, Dec 07 2022