A358882 - OEIS

%I #27 Dec 08 2022 08:53:33

%S 4,56,504,2024,8064,18200,50736,99248,202688,343256,657904,983008,

%T 1708672,2485968,3755184,5289944,8069736,10539792,15387320,19913840

%N The number of regions in a Farey diagram of order (n,n).

%C See A358298 and also the linked references for further details.

%C The first diagram where not all edge points are connected is n = 3. For example a line connecting points (0,1/3) and (1/3,0) has equation 3*y - 6*x - 1 = 0, and as one of the x or y coefficients is greater than n (3 in this case) the line is not included.

%H Alain Daurat, M. Tajine, M. Zouaoui, <a href="https://doi.org/10.1016/j.cag.2008.11.001">About the frequencies of some patterns in digital planes. Application to area estimators</a>. Computers & Graphics. 33.1 (2009), 11-20.

%H Daniel Khoshnoudirad, <a href="https://doi.org/10.2298/AADM150219008K">Farey lines defining Farey diagrams and application to some discrete structures</a>. Applicable Analysis and Discrete Mathematics. 9 (2015), 73-84.

%H Scott R. Shannon, <a href="/A358882/a358882.jpg">Image for n = 1</a>.

%H Scott R. Shannon, <a href="/A358882/a358882_1.jpg">Image for n = 2</a>.

%H Scott R. Shannon, <a href="/A358882/a358882_2.jpg">Image for n = 3</a>.

%H Scott R. Shannon, <a href="/A358882/a358882_3.jpg">Image for n = 4</a>.

%H Scott R. Shannon, <a href="/A358882/a358882_4.jpg">Image for n = 5</a>.

%H Scott R. Shannon, <a href="/A358882/a358882_5.jpg">Image for n = 6</a>.

%H Scott R. Shannon, <a href="/A358882/a358882_6.jpg">Image for n = 7</a>.

%H Scott R. Shannon, <a href="/A358882/a358882_7.jpg">Image for n = 8</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Farey_sequence">Farey sequence</a>.

%F a(n) = A358884(n) - A358883(n) + 1 by Euler's formula.

%Y Cf. A358883 (vertices), A358884 (edges), A358885 (k-gons), A006842, A006843, A005728, A358886.

%Y See A358298 for definition of Farey diagram Farey(m,n).

%Y The Farey Diagrams Farey(m,n) are studied in A358298-A358307 and A358882-A358885, the Completed Farey Diagrams of order (m,n) in A358886-A358889.

%K nonn,more

%O 1,1

%A _Scott R. Shannon_ and _N. J. A. Sloane_, Dec 05 2022