I watched a video by Mathologer about the Fibonacci sequence and the golden ratio `phi` (aka phi) recently, and found out some things.
In the Fibonacci sequence, each number is the sum of the previous two, F(n)=F(n-1)+F(n-2).
` 1,1,2,3,5,8,13,21… `
Powers of ` phi ` obey the same rule, the current power is the sum of the previous two powers.
That’s why three of them have the same fractional part:
` phi^-1: 0.618… `
` phi^0: 1 `
` phi^1: 1.618… `
` phi^2:2.618… `
` phi^0 = 1`, so ` phi = phi^-1 + 1 `, and ` phi^2 = phi + 1 `
In the video he showed an approximation of the Fibonacci sequence, proportional to ` phi^n `
` phi^n / sqrt 5 `
Since this approaches the Fibonacci sequence, which is an integer sequence, the fractional part of the formula approaches zero. That made sense to me.
He also showed the Lucas numbers, which also follow the pattern that each is the sum of the previous two, and which can be approximated by powers of ` phi ` .
` 1, 3, 4, 7, 11, 18… `
I thought it’s pretty cool that powers of ` phi ` approach integers like that. I’m not sure why it happens.
` phi ` can be found by solving the equation for ` phi^2 `, which is ` phi^2 = phi + 1 `, and taking the larger, positive solution.
` phi = (1 + sqrt 5) / 2 = 1.618… `
That involves the square root of 5, and I wondered if there would be similar patterns with numbers based on smaller irrational square roots, root 2 and root 3.
I thought they might be solutions of equations:
- G(n) = j G(n-1) + k G(n-2)
- x^2 = j x + k
for integers j and k.
I did find some others:
- ` j=2, k=1 `
` G(n) = 2 G(n-1) + G(n-2) `
solution of ` x^2 = 2x + 1 ` , is ` x = 1 + sqrt(2) ` , x is about 2.4142 - ` j=2, k=2 `
` G(n) = 2 G(n-1) + 2 G(n-2) `
solution of ` x^2 = 2x + 2 ` , is ` x = 1 + sqrt(3) ` , x is about 2.732 - ` j=4, k=4 `
` G(n) = 4 G(n-1) + 4 G(n-2) `
solution of ` x^2 = 4x + 4 ` , is ` x = 2 + 2 sqrt(2) ` , x is about 4.828 - ` j=4, k=-2 `
` G(n) = 4 G(n-1) - 2 G(n-2) `
solution of ` x^2 = 4x - 2 ` , is ` x = 2 + sqrt(2) ` , x is about 3.4142 - ` j=4, k=-1 `
` G(n) = 4 G(n-1) + 1 G(n-2) `
solution of ` x^2 = 4x - 1 ` , is ` x = 2 + sqrt(3) ` , x is about 3.732
Powers of these seem to converge towards being integers. As with power of the golden ratio ` phi `, I don’t know why.
References
The video was Phi and the TRIBONACCI monster,