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Search: id:A123726
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| A123726 |
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Denominators of fractional partial quotients appearing in a continued fraction for the power series Sum_{n>=0} x^(2^n - 1)/(n+1)^s. |
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+0
2
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| 1, 1, 4, 1, 9, 1, 4, 1, 16, 1, 4, 1, 9, 1, 4, 1, 25, 1, 4, 1, 9, 1, 4, 1, 16, 1, 4, 1, 9, 1, 4, 1, 36, 1, 4, 1, 9, 1, 4, 1, 16, 1, 4, 1, 9, 1, 4, 1, 25, 1, 4, 1, 9, 1, 4, 1, 16, 1, 4, 1, 9, 1, 4, 1, 49, 1, 4, 1, 9, 1, 4, 1, 16, 1, 4, 1, 9, 1, 4, 1, 25, 1, 4, 1, 9, 1, 4, 1, 16, 1, 4, 1, 9, 1, 4, 1, 36, 1, 4
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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A123725(n) = (A007814(n) + 2)*(-1)^A073089(n+1) are the numerators of the partial quotients.
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FORMULA
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a(n) = (A007814(n) + 1)^2 for n>=1, with a(0)=1, where A007814(n) is the exponent of the highest power of 2 dividing n.
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EXAMPLE
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Surprisingly, the following analog of the Riemann zeta function:
Z(x,s) = Sum_{n>=0} x^(2^n-1)/(n+1)^s = 1 + x/2^s + x^3/3^s +x^7/4^s+..
may be expressed by the continued fraction:
Z(x,s) = [1; f(1)^s/x, -f(2)^s/x, -f(3)^s/x,...,f(n)^s*(-1)^e(n)/x,...]
such that the (2^n-1)-th convergent = Sum_{k=0..n} x^(2^k-1)/(k+1)^s,
where f(n) = (b(n)+2)/(b(n)+1)^2 and e(n) = A073089(n+1) for n>=1,
and b(n) = A007814(n) the exponent of highest power of 2 dividing n.
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PROGRAM
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(PARI) {a(n)=denominator(subst(contfrac(sum(m=0, #binary(n), 1/x^(2^m-1)/(m+1)), n+3)[n+1], x, 1))} (PARI) /* a(n) = (A007814(n)+1)^2: */ {a(n)=if(n==0, 1, (valuation(n, 2)+1)^2)}
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CROSSREFS
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Cf. A123725 (numerators); A007814, A073089.
Sequence in context: A100235 A089072 A036177 this_sequence A138675 A080103 A091419
Adjacent sequences: A123723 A123724 A123725 this_sequence A123727 A123728 A123729
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KEYWORD
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cofr,frac,nonn
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Oct 12 2006
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