A278149 Triangle T(n, m) giving in row n the denominators of the fractions for the Farey dissection of order n.

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, 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, 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

Offset

1,1

Comments

For the numerators see A278148.

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

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.

References

G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 121.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Clarendon Press, Oxford, 2003, pp. 29 - 31.

Links

Table of n, a(n) for n=1..135.

Formula

T(1, 1) = 2 and for n>= 2: T(n, 1) = n + 1, T(n, A002088(n)) = n + 1 and for

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)))).

Example

The triangle T(n, m) begins:n\m 1  2  3  4  5  6  7  8  9 10 11 12 ...
1:  2
2:  3  3
3:  4  5  5  4
4:  5  7  5  5  7  5
5:  6  9  7  8  7  7  8  7  9  6
6:  7 11  9  7  8  7  7  8  7  9 11  7
...
n = 7: 8 13 11 9 11 10 8 12 9 9 12 8 10 11 9 11 13 8,
n = 8: 9 15 13 11 9 11 10 11 13 12 9 9 12 13 11 10 11 9 11 13 15 9,
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,
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.
........................................
For the fractions  A278148(n, m) / T(n,m) and the actual dissection intervals for n=5 see the examples for A278148.

Crossrefs

Cf. A002088, A005728, A015614, A278148.

Sequence in context: A078338 A338587 A338580 * A007306 A286544 A238690

Adjacent sequences: A278146 A278147 A278148 * A278150 A278151 A278152

Keyword

nonn,tabf,frac,easy

Author

Wolfdieter Lang, Nov 22 2016

Status

approved