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≈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=(2.732²)/2≈3.732
3/0.4174=(3.7913²)/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.

Answer 1

1 / 1 = 1

1 / 2 = 0.5

3 / 2 = 0.66666

3 / 5 = 0.6

5 / 8 = 0.625

8 / 13 = 0.61538

13 / 21 = 0.619

21 / 34 = 0.6176

34 / 55 = 0.61818

55 / 89 = 0.61797

 

The ratio between two consecutive Fibonacci numbers - F(n)/F(n+1) - approaches the Golden Mean: (1 + √5) / 2 or 1.61803398875......

Answer 2

The pattern you seem to be looking at is for every integer n, the number g you seek is such that

n/g = n+g

This gives g²+ng-n=0 or that g=(-n±√(n²+4n))/2

If you use the plus sign you get

n= 1     g=0.618033989

n=2      g=0.732050808

n=3      g=0.791287847

n=4      g=0.828427125

and so on

The Golden Ratio is such that 1/g=g/(1+g), not quite the same

 

Answer 3

Hi Tudor - 

 

I love this question and your curiosity, and wish I had an answer for you.  Instead, I wanted to share this interesting series of videos about Fibonacci and Lucas numbers in hopes it might fuel your creative energy.  This third video in the series speaks more to your question, but I found all of them of interest:  https://www.youtube.com/watch?v=14-NdQwKz9w.

 

I stumbled on these videos while pursing my interest in the golden ratio as used in photography, architecture, and art.  I love to see one subject intersect another like this.

 

I hope you'll find something in these videos to help you along your way.  Wishing you all the best,

 

Cindy