A358949 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).

3, 10, 148, 1111, 9568, 23770, 126187, 308401, 855145, 1521733, 4591405, 6831040

Offset: 1

Author

Scott R. Shannon and N. J. A. Sloane, Dec 07 2022

Keywords

• nonn
• more

Comments

The number of points along each edge is given by A005728(n).

Links

• 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.
• 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: Video, Slides, Updates. (Mentions this sequence.)
• Wikipedia, Farey sequence.

Crossrefs

Cf. A358948 (regions), A358950 (edges), A358951 (k-gons), A358887, A006842, A006843, A005728, A358882.

Formula

a(n) = A358950(n) - A358948(n) + 1 by Euler's formula.