I am not a mathematician (but a computer scientist) and stumbled across the following in the analysis of an algorithm (Berthold Vöcking: How Asymmetry Helps Load Balancing). The author gives Knuth: The Art of Computer Programming Vol. 3 as a source for the golden ratio - but I could not find anything about the given relationship between the $d$-ary Fibonacci numbers and a generalized golden ratio in it, but would like to have a closer look at this. Google didn't help much either - probably because other names are used.

Here are the definitions and the relationship from the article:

Define $d$-ary Fibonacci numbers as follows: for $k\leq 0$, $F_d(k) = 0$, $F_d(1) = 1,\,$ and for $k\geq 0,\;$ $$F_d(k) = \Sigma_{i=1}^{d} F_d(k-i)$$

Define $\Phi_d = \lim_{k \rightarrow \infty} \sqrt[k]{F_d(k)}$

Then, $F_2$ corresponds to the usual Fibonacci numbers and $\Phi_2$ to the golden ratio and $\Phi_d$ is a generalization.

Can you point me towards a proof?


Use the explicit solution: http://en.wikipedia.org/wiki/Recurrence_relation#Linear_homogeneous_recurrence_relations_with_constant_coefficients. Is a linear combination of exponentials.


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