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A359694 Irregular table read by rows: T(n,k) is the number of k-gons, k>=3, in a regular drawing of a complete bipartite graph where the vertex positions on each part equal the Farey series of order n.

%I A359694 #13 Feb 16 2025 08:34:04
%S A359694 2,10,2,70,24,218,160,4,1254,1068,148,16,2254,2414,252,26,10082,11760,
%T A359694 1980,266,12,21410,25958,5096,648,36,4,53422,68208,14360,1980,168,20,
%U A359694 86986,118922,24028,3056,248,12,0,2,255678,346676,84344,12774,1132,110,4,2,365674,493530,119820,18600,1624,112,4
%N A359694 Irregular table read by rows: T(n,k) is the number of k-gons, k>=3, in a regular drawing of a complete bipartite graph where the vertex positions on each part equal the Farey series of order n.
%C A359694 The number of vertices along each edge is A005728(n). No formula is known.
%C A359694 See A359692 for other images of the graph.
%H A359694 Scott R. Shannon, <a href="/A359694/a359694.jpg">Image for n = 7</a>.
%H A359694 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>.
%H A359694 Wikipedia, <a href="https://en.wikipedia.org/wiki/Farey_sequence">Farey sequence</a>.
%F A359694 Sum of row n = A359692(n).
%e A359694 The table begins:
%e A359694 2;
%e A359694 10, 2;
%e A359694 70, 24;
%e A359694 218, 160, 4;
%e A359694 1254, 1068, 148, 16;
%e A359694 2254, 2414, 252, 26;
%e A359694 10082, 11760, 1980, 266, 12;
%e A359694 21410, 25958, 5096, 648, 36, 4;
%e A359694 53422, 68208, 14360, 1980, 168, 20;
%e A359694 86986, 118922, 24028, 3056, 248, 12, 0, 2;
%e A359694 255678, 346676, 84344, 12774, 1132, 110, 4, 2;
%e A359694 365674, 493530, 119820, 18600, 1624, 112, 4;
%e A359694 917478, 1244492, 334096, 57080, 5700, 478, 16, 4;
%e A359694 1335398, 1862666, 495536, 82642, 8096, 676, 24, 6;
%e A359694 2107042, 2989864, 788340, 128378, 12536, 932, 52, 4;
%e A359694 3195474, 4557430, 1230300, 205352, 20516, 1664, 80, 4;
%e A359694 .
%e A359694 .
%Y A359694 Cf. A359690 (vertices), A359691 (crossings), A359692 (regions), A359693 (edges), A005728, A290131, A359653, A358886, A358882, A006842, A006843.
%K A359694 nonn,tabf
%O A359694 1,1
%A A359694 _Scott R. Shannon_ and _N. J. A. Sloane_, Jan 11 2023