A359977 - OEIS

%I #14 Jan 30 2023 10:34:01

%S 1,5,20,8,2,50,57,3,169,274,31,5,303,646,41,2,1,889,2011,179,21,2,

%T 1685,4025,388,33,4,3466,8283,925,67,7,5624,13442,1498,106,9,1,11896,

%U 27907,3718,354,30,2,16976,40100,5182,461,33,1,32506,73806,11249,1118,61,6,46187,104453,16380,1747,123,1,1

%N Irregular table read by rows: T(n,k) is the number of k-gons, k>=3, 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 See A359974 and A359975 for images of the triangle.

%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.

%F Sum of row n = A359975(n).

%e The table begins:

%e 1;

%e 5;

%e 20, 8, 2;

%e 50, 57, 3;

%e 169, 274, 31, 5;

%e 303, 646, 41, 2, 1;

%e 889, 2011, 179, 21, 2;

%e 1685, 4025, 388, 33, 4;

%e 3466, 8283, 925, 67, 7;

%e 5624, 13442, 1498, 106, 9, 1;

%e 11896, 27907, 3718, 354, 30, 2;

%e 16976, 40100, 5182, 461, 33, 1;

%e 32506, 73806, 11249, 1118, 61, 6;

%e 46187, 104453, 16380, 1747, 123, 1, 1;

%e 67117, 152534, 24159, 2511, 181, 10, 1;

%e 95276, 213798, 34962, 3824, 295, 21;

%e .

%e .

%Y Cf. A359974 (vertices), A359975 (regions), A359976 (edges), A005728, A359971, A359694, A358951, A358889.

%K nonn,tabf

%O 1,2

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