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The connections between the Fibonacci sequence and the golden ratio $\Phi:=\frac{1+\sqrt{5}}{2}$ are very well known. On the other hand, is there some (non-trivial) relation between the Fibonacci sequence and the silver ratio $\Phi_2:=1+\sqrt{2}$? It is also well known that $\Phi_2$ is closely related to the Pell sequence $P$ defined recursively as $P(0)=0$, $P(1)=1$, $P(n)=2P(n-1)+P(n-2)$ (for $n\geq 2$), but so far I have not been able to find any reference explaining some connection between $\Phi_2$ and the Fibonacci sequence (or between the Pell sequence and the Fibonacci sequence).

Paraphrased versions of my question go as follows:

Is there a (non-trivial) relation between the Fibonacci numbers and the Pell numbers?

Is there a relation between $\Phi$ and $\Phi_2$?

Is there an interesting relation between the Fibonacci numbers and $\sqrt{2}$?

dragoon
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  • You might want to examine the continued fractions for these numbers. – Mark Bennet Aug 05 '17 at 10:42
  • Both are Lucas sequences, see Ribenboim's New Book of Prime Number Records Ch. 2, IV or https://en.wikipedia.org/wiki/Lucas_sequence – gammatester Aug 05 '17 at 12:10
  • Pell sequence has only one perfect square $169=13^2$, excluding the trivial $0$ and $1$ just like Fibonacci sequence has only one perfect square, namely $144=12^2$

    But I can't prove none of the previous statements, albeit I know that the second one has been proved

     – Raffaele Aug 05 '17 at 21:27
  • Thank you all for all your comments. Indeed, these objects can be constructed from general concepts, like the Lucas Sequences as mentioned by @gammatester . I was wondering something like a nice equation that calculates $\sqrt{2}$ or $\Phi_2$ in terms of the Fibonacci sequence (e.g., $\Phi$ is calculated nicely as $\lim_{n\to\infty}\frac{F_{n+1}}{F_n}$ where $F_n$ is the $n$-th Fibonacci number). – dragoon Aug 06 '17 at 01:23
  • How do you define "non-trivial" in the context of the first question? https://en.wikipedia.org/wiki/Triviality_(mathematics) Can you give an example of a trivial relation between Fibonacci numbers and the Pell numbers? – James Arathoon Aug 12 '17 at 09:04
  • See my first and only question (so far) on MSE https://math.stackexchange.com/questions/2384587/analysis-of-convergence-properties-of-a-series-approximation-to-sqrtx-and There is a possible connecting relation w.r.t. to Catalan numbers. – James Arathoon Aug 12 '17 at 09:36
  • Thanks James for your contribution. I think I should change "non-trivial" by "interesting". For example, in terms of Fibonacci numbers, the silver ratio can be expressed as $F_1+\sqrt{F_3}$, but this may be uninteresting.

    The Catalan numbers provide a nice connection as well.

     – dragoon Aug 15 '17 at 02:31
  • @dragoon: You're welcome. There are lots of similar and analogous properties e.g. the sum of squares relationships are almost identical; but that is not what you asked for. Perhaps a separate question is needed to highlight as many analogous properties as readers know and this might help give more pointers in finding answers to your more difficult question. – James Arathoon Aug 15 '17 at 20:29
  • See my answer to this question https://math.stackexchange.com/questions/2408108/what-are-some-mathematically-interesting-computations-involving-matrices/2408142#2408142 – James Arathoon Aug 28 '17 at 00:30

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