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≈2.732

For 3: 3/0.7913≈3.7913

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

1/0.382≈2.618

2/0.536=(2.732^{2})/2≈3.732

3/0.4174=(3.7913^{2})/2≈4.7913

Another one would be:

(0.618)^{2}=1−0.618≈0.382(0.618)2=1−0.618≈0.382

(0.732)^{2}=2−2(0.732)≈0.536(0.732)2=2−2(0.732)≈0.536

(0.7913)^{2}=3−3∗(0.7913)≈0.6261(0.7913)2=3−3∗(0.7913)≈0.6261

And another one:

(0.618)3≈0.618−(0.618)2(0.618)3≈0.618−(0.618)2

(0.732)3≈2(0.732)−2(0.732)2(0.732)3≈2(0.732)−2(0.732)2

(0.7913)3≈3(0.7913)−3(0.7913)2(0.7913)3≈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 Fn=2(F_{n−1}+F_{n−2})Fn=2(_{Fn−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 Fn=3(F_{n−1}+F_{n−2})Fn=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.