Sum of Fibonacci(n)/Lucas(n)

[Jonathan Post]

Dear Max A.,

Excellent!

In January, when I can submit "made up" sequences again to OEIS, I'll be
sure to give your comment and email address, okay?

I wonder how often the numerator and denominator of A000045(i)/A000032(i)
have a factor in common, and reduce?

-- Jonathan Vos Post

On 12/7/06, Max A. <maxale at gmail.com> wrote:
>
> I've got the following asymptotic:
>
> SUM[i=1..n] A000045(i)/A000032(i) = n/sqrt(5) + O(1).
>
> Max
>
> On 12/6/06, Jonathan Post <jvospost3 at gmail.com> wrote:
> > Is there a seqfan who can tell me about this pair of sequences, which
> seems
> > not to be in OEIS?
> >
> >  a(n) = numerator(SUM[i=1..n]F(i)/L(i)) =
> > numerator(SUM[i=1..n]A000045(i)/A000032(i)).
> >  b(n) = denominator(SUM[i=1..n]F(i)/L(i)) =
> > numerator(SUM[i=1..n]A000045(i)/A000032(i)).
> >  The fractions, reduced to lowest terms, appear to begin:
> >  0/2, 1/1, 4/3, 11/6, 95/42, 1255/462, 4381/1386, 7662223/1889118,
> > 80819870/17946621, 3642636055/735811461, ...
> >
> >  Example: 0 +
> > 1+1/3+2/4+3/7+5/11+8/18+13/29+21/47+34/76+55/123
> >  = 1/1+ 1/19+ 1/4+ 1/1+ 1/13+ 1/2+ 1/4+ 1/1+ 1/6+ 1/6+ 1/1+ 1/85+ 1/1+
> 1/4+
> > 1/2
> >  since the continued fractions and their convergents may matter.
> >
> >  After all the Fibonacci and Lucas-related seqs I've investigated, often
> > with expert help from OEIS editors, I ought to see this clearly, but do
> not.
> > I'm not even clear on the asymptotics.
> >
> >  Thanks.
> >
> >  -- Jonathan Vos Post
> >
>
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