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A278149
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Triangle T(n, m) giving in row n the denominators of the fractions for the Farey dissection of order n.
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1
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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
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OFFSET
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1,1
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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FORMULA
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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)))).
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EXAMPLE
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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.
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For the fractions A278148(n, m) / T(n,m) and the actual dissection intervals for n=5 see the examples for A278148.
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CROSSREFS
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KEYWORD
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nonn,tabf,frac,easy
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AUTHOR
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STATUS
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approved
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