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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 equal the Farey series of order n.
5

%I #11 Jan 30 2023 10:32:40

%S 1,5,48,239,1798,3950,19953,46007,123338,213793,637960,930635,2361080,

%T 3542822,5736344

%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 equal the Farey series of order n.

%C The number of vertices along the shorter edges is A005728(n). No formula for a(n) is known. The sequence is inspired by the Farey fan; see A360042.

%H Scott R. Shannon, <a href="/A359969/a359969.jpg">Image for n = 2</a>.

%H Scott R. Shannon, <a href="/A359969/a359969_1.jpg">Image for n = 3</a>.

%H Scott R. Shannon, <a href="/A359969/a359969_2.jpg">Image for n = 4</a>.

%H Scott R. Shannon, <a href="/A359969/a359969_3.jpg">Image for n = 5</a>.

%H Scott R. Shannon, <a href="/A359969/a359969_4.jpg">Image for n = 6</a>.

%F a(n) = A359970(n) - A359968(n) + 1 by Euler's formula.

%Y Cf. A359968 (vertices), A359970 (edges), A359971 (k-gons), A005728, A360042, A359975, A359690, A358948, A358886.

%K nonn,more

%O 1,2

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