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%I #10 Nov 23 2016 12:50:26
%S 2,3,3,4,5,5,4,5,7,5,5,7,5,6,9,7,8,7,7,8,7,9,6,7,11,9,7,8,7,7,8,7,9,
%T 11,7,8,13,11,9,11,10,8,12,9,9,12,8,10,11,9,11,13,8,9,15,13,11,9,11,
%U 10,11,13,12,9,9,12,13,11,10,11,9,11,13,15,9,10,17,15,13,11,14,13,11,10,11,13,12,16,11,11,16,12,13,11,10,11,13,14,11,13,15,17,10,11,19,17,15,13,11,14,13,11,17,13,11,13,12,16,11,11,16,12,13,11,13,17,11,13,14,11,13,15,17,19,11
%N Triangle T(n, m) giving in row n the denominators of the fractions for the Farey dissection of order n.
%C For the numerators see A278148.
%C The length of row n is A002088(n) = A005728(n) - 1.
%C See A278148 for the definition of the Farey dissection of order n of the interval [1/(n+1), n/(n+1)] into A015614(n) intervals J(n,j) = [l(n,j), r(n,j)] with r(n,j) = l(n,j+1), for j=1..A015614(n), where the fractions l(n,j) and r(n,j) are given in a comment of A278148 in terms of three consecutive members of the Farey fraction sequence of order n.
%D G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 121.
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Clarendon Press, Oxford, 2003, pp. 29 - 31.
%F T(1, 1) = 2 and for n>= 2: T(n, 1) = n + 1, T(n, A002088(n)) = n + 1 and for
%F m = 2..(A002088(n) - 1): T(n, m) = denominator(l(n,m)) = denominator(p(n,m)/q(n,m) - 1/(q(n,m)*(q(n,m) + q(n,m-1)))).
%e The triangle T(n, m) begins:n\m 1 2 3 4 5 6 7 8 9 10 11 12 ...
%e 1: 2
%e 2: 3 3
%e 3: 4 5 5 4
%e 4: 5 7 5 5 7 5
%e 5: 6 9 7 8 7 7 8 7 9 6
%e 6: 7 11 9 7 8 7 7 8 7 9 11 7
%e ...
%e n = 7: 8 13 11 9 11 10 8 12 9 9 12 8 10 11 9 11 13 8,
%e n = 8: 9 15 13 11 9 11 10 11 13 12 9 9 12 13 11 10 11 9 11 13 15 9,
%e n = 9: 10 17 15 13 11 14 13 11 10 11 13 12 16 11 11 16 12 13 11 10 11 13 14 11 13 15 17 10,
%e n = 10: 11 19 17 15 13 11 14 13 11 17 13 11 13 12 16 11 11 16 12 13 11 13 17 11 13 14 11 13 15 17 19 11.
%e ........................................
%e For the fractions A278148(n, m) / T(n,m) and the actual dissection intervals for n=5 see the examples for A278148.
%Y Cf. A002088, A005728, A015614, A278148.
%K nonn,tabf,frac,easy
%O 1,1
%A _Wolfdieter Lang_, Nov 22 2016
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