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Number of regions 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 #14 Jan 30 2023 10:33:44

%S 1,5,30,110,479,993,3102,6135,12748,20680,43907,62753,118746,168892,

%T 246513,348176,571980,725956,1129035,1426393,1887096,2387945,3454566,

%U 4123548,5543837

%N Number of regions 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="/A359975/a359975.jpg">Image for n = 2</a>.

%H Scott R. Shannon, <a href="/A359975/a359975_1.jpg">Image for n = 3</a>.

%H Scott R. Shannon, <a href="/A359975/a359975_2.jpg">Image for n = 4</a>.

%H Scott R. Shannon, <a href="/A359975/a359975_3.jpg">Image for n = 5</a>.

%H Scott R. Shannon, <a href="/A359975/a359975_4.jpg">Image for n = 6</a>.

%F a(n) = A359976(n) - A359974(n) + 1 by Euler's formula.

%Y Cf. A359974 (vertices), A359976 (edges), A359977 (k-gons), A005728, A359969, A359690, A358948, A358886.

%K nonn,more

%O 1,2

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