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Number of vertices in a regular drawing of a complete bipartite graph where the vertex positions on each part equal the Farey series of order n.
10

%I #13 Jan 12 2023 16:12:08

%S 5,13,69,289,1971,3997,20371,45751,120957,205299,629847,897801,

%T 2334409,3461459,5517131,8468061

%N Number of vertices in a regular drawing of a complete bipartite graph where the vertex positions on each part equal the Farey series of order n.

%C The number of vertices along each edge is A005728(n). No formula for a(n) is known.

%H Scott R. Shannon, <a href="/A359690/a359690.png">Image for n = 1</a>.

%H Scott R. Shannon, <a href="/A359690/a359690_1.png">Image for n = 2</a>.

%H Scott R. Shannon, <a href="/A359690/a359690_2.png">Image for n = 3</a>.

%H Scott R. Shannon, <a href="/A359690/a359690_3.png">Image for n = 4</a>.

%H Scott R. Shannon, <a href="/A359690/a359690_4.png">Image for n = 5</a>.

%H Scott R. Shannon, <a href="/A359690/a359690_5.png">Image for n = 6</a>.

%H Scott R. Shannon, <a href="/A359690/a359690_6.png">Image for n = 7</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>.

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

%F a(n) = A359693(n) - A359692(n) + 1 by Euler's formula.

%Y Cf. A359691 (crossings), A359692 (regions), A359693 (edges), A359694 (k-gons), A005728, A331755, A359654, A358887, A358883, A006842, A006843.

%K nonn,more

%O 1,1

%A _Scott R. Shannon_ and _N. J. A. Sloane_, Jan 11 2023