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%I #11 Sep 27 2023 14:55:53
%S 1,12,228,1464,12516,29022,153564,364650,996672,1750326,5274156,
%T 7761498
%N Number of regions 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="/A358948/a358948.jpg">Image for n = 2</a>.
%H Scott R. Shannon, <a href="/A358948/a358948_1.jpg">Image for n = 3</a>.
%H Scott R. Shannon, <a href="/A358948/a358948_2.jpg">Image for n = 4</a>.
%H Scott R. Shannon, <a href="/A358948/a358948_3.jpg">Image for n = 5</a>.
%H Scott R. Shannon, <a href="/A358948/a358948_4.jpg">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) - A358949(n) + 1 by Euler's formula.
%Y Cf. A358949 (vertices), A358950 (edges), A358951 (k-gons), A358886, A006842, A006843, A005728, A358882.
%K nonn,more
%O 1,2
%A _Scott R. Shannon_ and _N. J. A. Sloane_, Dec 07 2022