A360042 Number of vertices in a Farey fan of order n.

4, 6, 11, 17, 29, 39, 59, 79, 107, 133, 175, 213, 271, 323, 385, 451, 541, 621, 731, 835, 955, 1073, 1225, 1367, 1541, 1707, 1897, 2087, 2321, 2535, 2801, 3061, 3345, 3625, 3937, 4243, 4609, 4957, 5335, 5713, 6155, 6569, 7055, 7529, 8031, 8531, 9101, 9649, 10265, 10859

Offset: 1

Author

Scott R. Shannon, N. J. A. Sloane and M. Douglas McIlroy Jan 23 2023

Keywords

• nonn

Comments

See the reference for the definition of a 'Farey fan'.

The number of vertices along each edge is A005728(n), while the number of regions is conjectured to equal A005598(n) = 1 + Sum_{i=1..n} (n-i+1)*phi(i). The regions count the number of distinct approximate representations of straight lines y = mx + b that can be drawn on an x-y integer raster, where x, y, and b are restricted to [0,n) and 0 <= m <=1.

It is also worth noting that for 3 <= n <= 10 this sequence equals 2*A005728(n) + A174030(n-2), where A174030(n) = Sum_{i=1..n} (i where phi(i)|i). That is, the number of internal vertices of the Farey fan equals A174030(n) in this range. This may suggest a possible attack on finding a formula for the present sequence.

Links

• M. Douglas McIlroy, A Note on Discrete Representation of Lines, AT&T Technical Journal, 64 (1985), 481-490.
• 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.
• Scott R. Shannon, Image for n = 10.

Crossrefs

Cf. A005598 (regions), A360043 (edges), A360044 (k-gons), A005728, A174030, A359974, A359968, A359690.