I'm not very mathematical, so yea, sorry... I know that the ratio when dividing any two fibonacci numbers approaches the golden ratio. Is there any way to find what the ratio when dividing two generalised fibonacci sequences approaches?
I have two sequences, $h$ and $k$, where $$ h_{n}=(n+1)h_{n-1} + h_{n-2}\\ k_{n}=(n+1)k_{n-1} + k_{n-2} $$
$ h_{0}=0, h_{1} = 1\\ k_{0}=1, k_{1}=2 $
and I need $\lim\limits_{n \to +\infty} \frac{h_n}{k_n}$
So approximations start as $ \frac{1}{2}, \frac{3}{7}, \frac{13}{30} $ etc etc. I'm not sure if this helps, but when searching some of those approximations, one of them flagged up an integer series, the approximation $\frac{421}{972}$ showed me this: https://oeis.org/A058294 The following approximation in the sequence was also there, $\frac{3015}{6961}$
Alternatively, I know my problem could be solved if I could evaluate $$ \sum_{n=0}^{\infty} \frac{(-1)^n}{k_n \times k_{n+1}} $$ But I'm guessing the 1st would be much easier, just tell me if I'm wrong :) thanks.
Okay, I've solved it thanks a legend in a discord server :) The limit is equivalent to
$$ \frac{I_2(2)}{I_1(2)} $$
Which itself is equivalent to
$$ \frac{2}{3} \times \frac{ \int_{0}^\pi e^{-2cos(t)} sin^4(t) dt }{ \int_{0}^\pi e^{-2cos(t)} sin^2(t) dt } $$
Where $I_n$ is a modified Bessel function.