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A278148 Triangle T(n, m) giving in row n the numerators of the fractions for the Farey dissection of order n.

%I A278148 #17 Apr 01 2017 20:22:50
%S A278148 1,1,2,1,2,3,3,1,2,2,3,5,4,1,2,2,3,3,4,5,5,7,5,1,2,2,2,3,3,4,5,5,7,9,
%T A278148 6,1,2,2,2,3,3,3,5,4,5,7,5,7,8,7,9,11,7,1,2,2,2,2,3,3,4,5,5,4,5,7,8,7,
%U A278148 7,8,7,9,11,13,8,1,2,2,2,2,3,3,3,3,4,5,5,7,5,6,9,7,8,7,7,8,10,11,9,11,13,15,9
%N A278148 Triangle T(n, m) giving in row n the numerators of the fractions for the Farey dissection of order n.
%C A278148 For the denominators see A278149.
%C A278148 The length of row n is A002088(n) = A005728(n) - 1.
%C A278148 In the Hardy reference one finds from the Farey fractions of order n >= 2 (see A006842/A006843) a dissection of the interval [1/(n+1), n/(n+1)] into A015614(n) = A005728(n) - 2  intervals J(n,j) = [l(n,j), r(n,j)], j = 1..A015614(n). They are obtained from three consecutive Farey fractions of order n: p(n,j-1)/q(n,j-1), p(n,j)/q(n,j), p(n,j+1)/q(n,j+1) by l(n,j) = p(n,j)/q(n,j) - 1/(q(n,j)*(q(n,j) + q(n,j-1))) = (p(n,j) + p(n,j-1))/(q(n,j) + q(n,j-1)) and r(n,j) = p(n,j)/q(n,j) + 1/(q(n,j)*(q(n,j) + q(n,j+1))) = (p(n,j) + p(n,j+1))/(q(n,j) + q(n,j+1)). (Hardy uses N for n, p/q - Chi_{p,q}'' for l (left) and p/q + Chi_{p,q}' for r (right)). For the second equations in l(n,j) and r(n,j) see the identities in Hardy-Wright, p. 23, Theorem 28.
%C A278148 Due to r(n,j) = l(n,j+1), for n >= 2 and j=1..A015614(n), it is sufficient to give for this Farey dissection of order n >= 2 only the two endpoints of the interval [1/(n+1), n/(n+1)] and the A015614(n) - 1 = A002088 - 2 = A005728(n) - 3 inner boundary points. The present table gives the numerators of these fractions. See the example section. For n = 1 we add the row 1/2 in accordance with A002088(1) = 1.
%D A278148 G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 121.
%D A278148 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Clarendon Press, Oxford, 2003, pp. 23, 29 - 31.
%F A278148 T(1, 1) = 1 and for n>= 2:  T(n, 1) = 1, T(n, A002088(n)) = n and for m = 2..(A002088(n) - 1): T(n, m) = numerator(l(n,m)) = numerator( p(n,m)/q(n,m) - 1/(q(n,m)*(q(n,m) + q(n,m-1)))).
%e A278148 The triangle T(n, m) begins:
%e A278148 n\m 1 2 3 4 5 6 7 8 9 10 11 12 ...
%e A278148 1:  1
%e A278148 2:  1 2
%e A278148 3:  1 2 3 3
%e A278148 4:  1 2 2 3 5 4
%e A278148 5:  1 2 2 3 3 4 5 5 7  5
%e A278148 6:  1 2 2 2 3 3 4 5 5  7  9  6
%e A278148 ...
%e A278148 n = 7:  1 2 2 2 3 3 3 5 4 5 7 5 7 8 7 9 11 7,
%e A278148 n = 8:  1 2 2 2 2 3 3 4 5 5 4 5 7 8 7 7 8 7 9 11 13 8,
%e A278148 n = 9:  1 2 2 2 2 3 3 3 3 4 5 5 7 5 6 9 7 8 7 7 8 10 11 9 11 13 15 9,
%e A278148 n = 10: 1 2 2 2 2 2 3 3 3 5 4 4 5 5 7 5 6 9 7 8 7 9 12 8 10 11 9 11 13 15 17 10.
%e A278148 .............................................
%e A278148 The fractions T(n,m)/A278149(n, m) begin:
%e A278148 n\m 1    2   3   4   5   6   7   8   9  10
%e A278148 1: 1/2
%e A278148 2: 1/3  2/3
%e A278148 3: 1/4  2/5 3/5 3/4
%e A278148 4: 1/5  2/7 2/5 3/5 5/7 4/5
%e A278148 5: 1/6  2/9 2/7 3/8 3/7 4/7 5/8 5/7 7/9 5/6
%e A278148 ...
%e A278148 n = 6: 1/7 2/11 2/9 2/7 3/8 3/7 4/7 5/8 5/7 7/9 9/11 6/7,
%e A278148 n = 7: 1/8 2/13 2/11 2/9 3/11 3/10 3/8 5/12 4/9 5/9 7/12 5/8 7/10 8/11 7/9 9/11 11/13 7/8,
%e A278148 n = 8: 1/9 2/15 2/13 2/11 2/9 3/11 3/10 4/11 5/13 5/12 4/9 5/9 7/12 8/13 7/11 7/10 8/11 7/9 9/11 11/13 13/15 8/9,
%e A278148 n = 9: 1/10 2/17 2/15 2/13 2/11 3/14 3/13 3/11 3/10 4/11 5/13 5/12 7/16 5/11 6/11 9/16 7/12 8/13 7/11 7/10 8/11 10/13 11/14 9/11 11/13 13/15 15/17 9/10,
%e A278148 n = 10: 1/11 2/19 2/17 2/15 2/13 2/11 3/14 3/13 3/11 5/17 4/13 4/11 5/13 5/12 7/16 5/11 6/11 9/16 7/12 8/13 7/11 9/13 12/17 8/11 10/13 11/14 9/11 11/13 13/15 15/17 17/19 10/11.
%e A278148 .............................................
%e A278148 For n = 5 the actual intervals J(5,j), j= 1..9 are then:
%e A278148 [1/6, 2/9], [2/9, 2/7], [2/7, 3/8], [3/8, 3/7], [3/7, 4/7], [4/7, 5/8], [5/8, 5/7], [5/7, 7/9], [7/9, 5/6].
%Y A278148 Cf. A002088, A005728, A006842/A006843, A015614, A278149.
%K A278148 nonn,tabf,frac,easy
%O A278148 1,3
%A A278148 _Wolfdieter Lang_, Nov 22 2016