Sum of Fibonacci(n)/Lucas(n)

[Max A.]

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
>