A358948 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).

1, 12, 228, 1464, 12516, 29022, 153564, 364650, 996672, 1750326, 5274156, 7761498

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..12.

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.

Formula

a(n) = A358950(n) - A358949(n) + 1 by Euler's formula.

Crossrefs

Cf. A358949 (vertices), A358950 (edges), A358951 (k-gons), A358886, A006842, A006843, A005728, A358882.

Sequence in context: A098647 A357410 A300561 * A361184 A358362 A337332

Adjacent sequences: A358945 A358946 A358947 * A358949 A358950 A358951

Keyword

nonn,more

Author

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

Status

approved