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A123725
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Numerators 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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3
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1, 2, -3, -2, -4, 2, 3, -2, -5, 2, -3, -2, 4, 2, 3, -2, -6, 2, -3, -2, -4, 2, 3, -2, 5, 2, -3, -2, 4, 2, 3, -2, -7, 2, -3, -2, -4, 2, 3, -2, -5, 2, -3, -2, 4, 2, 3, -2, 6, 2, -3, -2, -4, 2, 3, -2, 5, 2, -3, -2, 4, 2, 3, -2, -8, 2, -3, -2, -4, 2, 3, -2, -5, 2, -3, -2, 4, 2, 3, -2, -6, 2, -3, -2, -4, 2, 3, -2, 5, 2, -3, -2, 4, 2, 3, -2, 7, 2, -3, -2, -4, 2
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OFFSET
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0,2
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LINKS
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Antti Karttunen, Table of n, a(n) for n = 0..65537
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FORMULA
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a(n) = (A007814(n) + 2) * (-1)^A073089(n+1), n>=1, with a(0)=1.
a(n) = A089080(n+1) * (-1)^A073089(n+1) for n>=0.
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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.
Thus a(n) = (A007814(n) + 2)*(-1)^A073089(n+1) are numerators and
A123726(n) = (A007814(n) + 1)^2 are denominators of partial quotients.
Case s=1.
Sum_{n>=0} x^(2^n-1)/(n+1) = [1; 2/x, -(3/4)/x, -2/x, -(4/9)/x, 2/x,
(3/4)/x, -2/x, -(5/16)/x, 2/x, -(3/4)/x, -2/x, (4/9)/x, 2/x, (3/4)/x,
-2/x, -(6/25)/x, 2/x, -(3/4)/x, -2/x, -(4/9)/x, 2/x, (3/4)/x, -2/x,...].
Note that (2^n-1)-th convergents exactly equal n-th partial sums:
[1;2/x] = 1 + x/2;
[1;2/x,-(3/4)/x,-2/x] = 1 + x/2 + x^3/3;
[1;2/x,-(3/4)/x,-2/x,-(4/9)/x,2/x,(3/4)/x,-2/x] = 1 +x/2 +x^3/3 +x^7/4.
Case s=2.
Sum_{n>=0} x^(2^n-1)/(n+1)^2 = [1; 4/x, -(9/16)/x, -4/x, -(16/81)/x,
4/x, (9/16)/x, -4/x, -(25/256)/x, 4/x, -(9/16)/x, -4/x, (16/81)/x, 4/x,
(9/16)/x, -4/x, -(36/625)/x, 4/x, -(9/16)/x, -4/x, (16/81)/x, 4/x ...].
Note that (2^n-1)-th convergents exactly equal n-th partial sums:
[1;4/x] = 1 + x/4;
[1;4/x,-(9/16)/x,-4/x] = 1 + x/4 + x^3/9;
[1;4/x,-(9/16)/x,-4/x,-(16/81)/x,4/x,(9/16)/x,-4/x]=1+x/4+x^3/9+x^7/16.
Case s=3.
Sum_{n>=0} x^(2^n-1)/(n+1)^3 = [1; 8/x,-(27/64)/x,-8/x,-(64/729)/x,8/x,
(27/64)/x,-8/x,-(125/4096)/x, 8/x,-(27/64)/x,-8/x, (64/729)/x, 8/x,
(27/64)/x,-8/x,-(216/15625)/x, 8/x, -(27/64)/x, -8/x, (64/729)/x ...].
Likewise, the (2^n-1)-th convergents exactly equal n-th partial sums.
It is conjectured that these patterns continue for all s.
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PROG
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(PARI) {a(n)=numerator(subst(contfrac(sum(m=0, #binary(n), 1/x^(2^m-1)/(m+1)), n+3)[n+1], x, 1))}
(PARI)
A007814(n) = valuation(n, 2);
A073089(n) = { if(n<=1, return(0)); n-=1; my(v=2^valuation(n, 2)); return((0==bitand(n, v<<1)) != (v%2)); }; \\ From A073089
A123725(n) = if(!n, 1, (A007814(n) + 2) * (-1)^A073089(n+1)); \\ Antti Karttunen, Nov 01 2018
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CROSSREFS
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Cf. A123726 (denominators); A007814, A073089, A089080 (unsigned).
Sequence in context: A199968 A066482 A324507 * A089080 A085058 A183152
Adjacent sequences: A123722 A123723 A123724 * A123726 A123727 A123728
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KEYWORD
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cofr,frac,sign
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AUTHOR
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Paul D. Hanna, Oct 12 2006
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STATUS
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approved
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