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A358885 Table read by rows: T(n,k) = the number of regions with k sides, k >= 3, in a Farey diagram of order (n,n).

%I A358885 #19 Dec 06 2022 19:33:37
%S A358885 4,48,8,400,104,1568,456,6216,1848,13944,4256,38760,11976,75768,23480,
%T A358885 154440,48248,261072,82184,500464,157440,747480,235528,1298584,410088,
%U A358885 1890184,595784,2853416,901768,4015552,1274392,6127632,1942104,8002552,2537240,11683880,3703440,15123800,4790040
%N A358885 Table read by rows: T(n,k) = the number of regions with k sides, k >= 3, in a Farey diagram of order (n,n).
%C A358885 See the linked references for further details.
%C A358885 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.
%C A358885 It would be nice to have a proof (or disproof) that the number of sides is always 3 or 4.
%H A358885 Alain Daurat et al., <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 A358885 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 A358885 Scott R. Shannon, <a href="/A358885/a358885.jpg">Image for n = 5</a>.
%H A358885 Wikipedia, <a href="https://en.wikipedia.org/wiki/Farey_sequence">Farey sequence</a>.
%F A358885 Sum of row n = A358882(n).
%e A358885 The table begins:
%e A358885 4;
%e A358885 48, 8;
%e A358885 400, 104;
%e A358885 1568, 456;
%e A358885 6216, 1848;
%e A358885 13944, 4256;
%e A358885 38760, 11976;
%e A358885 75768, 23480;
%e A358885 154440, 48248;
%e A358885 261072, 82184;
%e A358885 500464, 157440;
%e A358885 747480, 235528;
%e A358885 1298584, 410088;
%e A358885 1890184, 595784;
%e A358885 2853416, 901768;
%e A358885 4015552, 1274392;
%e A358885 6127632, 1942104;
%e A358885 8002552, 2537240;
%e A358885 11683880, 3703440;
%e A358885 15123800, 4790040;
%e A358885 .
%e A358885 .
%Y A358885 Cf.  A358882 (regions), A358883 (vertices), A358884 (edges), A006842, A006843, A005728, A358889.
%Y A358885 See A358298 for definition of Farey diagram Farey(m,n).
%Y A358885 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 A358885 nonn,tabf
%O A358885 1,1
%A A358885 _Scott R. Shannon_ and _N. J. A. Sloane_, Dec 05 2022