A360042 - OEIS

%I #18 Jan 30 2023 10:34:33

%S 4,6,11,17,29,39,59,79,107,133,175,213,271,323,385,451,541,621,731,

%T 835,955,1073,1225,1367,1541,1707,1897,2087,2321,2535,2801,3061,3345,

%U 3625,3937,4243,4609,4957,5335,5713,6155,6569,7055,7529,8031,8531,9101,9649,10265,10859

%N Number of vertices in a Farey fan of order n.

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

%C 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.

%C 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.

%H M. Douglas McIlroy, <a href="https://doi.org/10.1002/j.1538-7305.1985.tb00359.x">A Note on Discrete Representation of Lines</a>, AT&T Technical Journal, 64 (1985), 481-490.

%H Scott R. Shannon, <a href="/A360042/a360042.png">Image for n = 2</a>.

%H Scott R. Shannon, <a href="/A360042/a360042_1.png">Image for n = 3</a>.

%H Scott R. Shannon, <a href="/A360042/a360042_2.png">Image for n = 4</a>.

%H Scott R. Shannon, <a href="/A360042/a360042_3.png">Image for n = 5</a>.

%H Scott R. Shannon, <a href="/A360042/a360042_4.png">Image for n = 6</a>.

%H Scott R. Shannon, <a href="/A360042/a360042_5.png">Image for n = 10</a>.

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

%K nonn

%O 1,1

%A _Scott R. Shannon_, _N. J. A. Sloane_ and _M. Douglas McIlroy_ Jan 23 2023