Fibonacci: other golden ratios for higher numbers?

Question

I just want to know what this property that I'm about to show is called. So I'm not a mathematician and this may sound dumb :) but I was looking over a Fibonacci article yesterday and studied it's properties and I wondered: Well, if 1 has the golden ratio of 0.618 what if 2 has a certain golden ratio and 3 has one. And through trial and error I came up with this numbers that have the same properties analogous to the 1 and 0.618. For example:

$1/0.618 = 1.618$

For 2 I found: $2/0.732 \approx 2.732$

For 3: $3/0.7913 \approx 3.7913$

Now what I found was that these numbers share similar properties. For example:

$1/0.382 \approx 2.618$

$2/0.536 = \frac {2.732^2}{2} \approx 3.732$

$3/0.4174 = \frac {3.7913^2}{2} \approx 4.7913$

Another one would be:

$(0.618)^2 = 1- 0.618 \approx 0.382$

$(0.732)^2 = 2- 2(0.732) \approx 0.536$

$(0.7913)^2 = 3-3*(0.7913) \approx 0.6261$

And another one:

$(0.618)^3 \approx 0.618 - (0.618)^2$

$(0.732)^3 \approx 2(0.732) - 2(0.732)^2$

$(0.7913)^3 \approx 3(0.7913) - 3(0.7913)^2$

I didn't test for other properties cause I think it's enough to make a point. So then I thought that there has to be a string of numbers to have same properties as the Fibonacci with 0.618 ratio.

So I realized that for:

0.732 it's 2 4 12 32 88 240 656 1792... Basically $F_n = 2(F_{n-1} + F_{n-2})$

And if you divide 1792/656 you get 2.732. If you divide 656/1792 you get approx 0.366 which times 2 is 0.732.

Same for 0.7913 it's 3 9 36 135 513 1944 7371 formula being $F_n = 3(F_{n-1} + F_{n-2})$

And same if you divide 7371/1944 you get approx 3.7913. And if 1944/7371 you get approx 0.2637 which times 3 is approx 0.7913.

Now what I want to know is what are these numbers or this property of numbers called? I looked for these ratios but didn't find anything.

Answer 1

Well, I don't know if there is a name for it, but it isn't that much mysterious.

If $k=1,2$ or $3$ in your setting (it could be any $k\in\mathbb{N}^*$), you are computing the positive solution of

$$\frac{k}{x}=k+x$$

which for example give $x\sim 0.618$ if $k=1$. This number can be explicitly written in terms of $k$, if you know how to solve a polynomial equation of degree 2 ($x^2+k.x-k=0$).

From there, the properties you noticed are reasonably easy to check: $x^2=k-k.x$, $x^3=x.x^2=k.x-k.x^2$.

As for the ratio, if you are indeed interested in the numbers satisfying $F_n=k.(F_{n-1}+F_{n-2})$, then denoting $u_n=\frac{F_{n-1}}{F_n}$, you have $$\frac{1}{u_n}=k+k.u_{n-1},$$ so if $u_n$ has a limit (which remains to be proven), which is denoted $\ell$, then you indeed get: $$\frac{1}{\ell}=k+k.\ell$$ and if you define $x=k.\ell$, you indeed obtain $$\frac{k}{x}=\frac{1}{\ell}=k+k.\ell=k+x.$$

My point is that I don't believe that these numbers enjoys enough remarkable properties to be honoured to have a name. The gold number is very famous, because the Fibonacci sequence was very famous, but also because this ratio appears in many areas others than mathematics. But for people knowing a bit about modern mathematics, the gold number is not so peculiar, and can be easily written just using fractions and square roots.

Answer 2

$\displaystyle{2+\frac{1}{2+\frac{1}{2+\ldots}}}$ can be called silver ratio;

$\displaystyle{3+\frac{1}{3+\frac{1}{3+\ldots}}}$ can be called bronze ratio;

See also https://en.wikipedia.org/wiki/Metallic_mean.