A278148 - OEIS

%I #17 Apr 01 2017 20:22:50

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

%C For the denominators see A278149.

%C The length of row n is A002088(n) = A005728(n) - 1.

%C 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 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 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. 23, 29 - 31.

%F 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 The triangle T(n, m) begins:

%e n\m 1 2 3 4 5 6 7 8 9 10 11 12 ...

%e 1:  1

%e 2:  1 2

%e 3:  1 2 3 3

%e 4:  1 2 2 3 5 4

%e 5:  1 2 2 3 3 4 5 5 7  5

%e 6:  1 2 2 2 3 3 4 5 5  7  9  6

%e ...

%e n = 7:  1 2 2 2 3 3 3 5 4 5 7 5 7 8 7 9 11 7,

%e 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 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 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 .............................................

%e The fractions T(n,m)/A278149(n, m) begin:

%e n\m 1    2   3   4   5   6   7   8   9  10

%e 1: 1/2

%e 2: 1/3  2/3

%e 3: 1/4  2/5 3/5 3/4

%e 4: 1/5  2/7 2/5 3/5 5/7 4/5

%e 5: 1/6  2/9 2/7 3/8 3/7 4/7 5/8 5/7 7/9 5/6

%e ...

%e 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 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 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 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 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 .............................................

%e For n = 5 the actual intervals J(5,j), j= 1..9 are then:

%e [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 Cf. A002088, A005728, A006842/A006843, A015614, A278149.

%K nonn,tabf,frac,easy

%O 1,3

%A _Wolfdieter Lang_, Nov 22 2016