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Number of vertices formed inside a right triangle by the straight line segments mutually connecting all vertices and points on the two shorter edges whose positions on one edge equal the Farey series of order n while on the other they divide its length into n equal segments.
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%I #13 Jan 30 2023 10:33:36

%S 3,6,26,93,424,876,2785,5542,11575,18761,40249,57399,109376,155965,

%T 227884,322377,532454,676282,1056010,1334975,1767798,2240664,3252047,

%U 3882192,5226897

%N Number of vertices formed inside a right triangle by the straight line segments mutually connecting all vertices and points on the two shorter edges whose positions on one edge equal the Farey series of order n while on the other they divide its length into n equal segments.

%C The number of vertices on the edge with point positions equaling the Farey series of order n is A005728(n). No formula for a(n) is known.

%C This graph is related to the 'Farey fan' given in the reference.

%D McIlroy, M. D. "A Note on Discrete Representation of Lines". AT&T Technical Journal, 64 (1985), 481-490.

%H Scott R. Shannon, <a href="/A359974/a359974.png">Image for n = 2</a>.

%H Scott R. Shannon, <a href="/A359974/a359974_1.png">Image for n = 3</a>.

%H Scott R. Shannon, <a href="/A359974/a359974_2.png">Image for n = 4</a>.

%H Scott R. Shannon, <a href="/A359974/a359974_3.png">Image for n = 5</a>.

%H Scott R. Shannon, <a href="/A359974/a359974_4.png">Image for n = 6</a>.

%F a(n) = A359976(n) - A359975(n) + 1 by Euler's formula.

%Y Cf. A359975 (regions), A359976 (edges), A359977 (k-gons), A005728, A359968, A359690, A358949, A358887.

%K nonn,more

%O 1,1

%A _Scott R. Shannon_ and _N. J. A. Sloane_, Jan 20 2023