Fibonacci? Too mainstream..

So I really enjoy infinite sequences that correspond to a constant value (most cool people do) and I wanted to post about a couple including a recent favourite the Look-and-say Sequence. The Fibonacci Sequence is probably the most known. Every Nth number (after the first two) is the sum of the previous two:

F_n = F_{n-1} + F_{n-2} where F_0 = 0 and F_1 = 1
\boldsymbol{F} = ( 0,1,2,3,5,7,13,21,34,55,89,\dots )

The constant value that corresponds to this sequence is called the Golden Ratio, which I use in this previous post. The sequence converges to the ratio:

\displaystyle \lim_{n\to\infty}\frac{F_{n+1}}{F_n}=\varphi \approx 1.61803

The Lucas Numbers:
Changing the starting value of the Fibonacci sequence from 0 to 2 gives:

L_n = L_{n-1} + L_{n-2} where L_0 = 2 and L_1 = 1
\boldsymbol{L} = ( 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123,\dots )

The Lucas Numbers relate directly to the Fibonaci Numbers, for example: L_n = F_{n-1} + F_{n+1} and even though almost all the numbers are different, they both converge to the same ratio:

\displaystyle \lim_{n\to\infty}\frac{L_{n+1}}{L_n}=\varphi \approx 1.61803

The Pell Numbers:
With a slight alteration (multiplying one term by 2) the Fibonacci equation produces:

P_n = 2 P_{n-1} + P_{n-2} where P_0 = 0 and P_1 = 1
\boldsymbol{P} = ( 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378,\dots )

The Pell numbers related to \sqrt{2} which can be seen when the series converges to the Silver Ratio (1+\sqrt{2}):

\displaystyle \lim_{n\to\infty}\frac{P_{n+1}}{P_n}=\delta_s \approx 2.1412

The Look-and-say Sequence:
This is a very interesting sequence because of it’s strange method. Take a ‘seed’ number (the first in the sequence), write it down then say what you see, repeat. For example if you start with 1 then that is “One 1”, written down that is “Two 1s”, written down that is “One 2, One 1”, etc. In the sequence no number greater than 3 is needed unless that number is used as the seed:

\boldsymbol{Ls} = ( 1,11,21,1211,111221,312211, 13112221, 1113213211, \dots )

If the seed number is 22 then nothing ever changes, otherwise the numbers eventually grow at a rate of 30% in size each time. Just as before a ratio emerges from the sequence and that gives us Conway’s Constant:

\displaystyle \lim_{n \to \infty}\frac{{Ls}_{n+1}}{{Ls}_{n}} = \lambda \approx 1.30357

This ratio is found with any seed number that isn’t 22. On a side note, the Golden Ratio is the root to the 2nd degree polynomial x^2 - x - 1 but Conway’s Constant isn’t as simple an equation, it is the root to a 71st degree polynomial! It also has links to chemistry and was first published about in a paper called “The Weird and Wonderful Chemistry of Audioactive Decay”.

The Super Fibonacci Numbers:
I can’t find anything on this so i’m not sure if they have a name or if I’ll be the first to write about them. I started with a number: The Super Golden Ratio (Sg) which is the solution to x^3 - x^2 - 1 = 0 and has the value of 1.4655\dots
Using the closed-form expression called Binet’s Formula I could easily get the set of numbers, all I had to do was swap the golden ratio and golden conjugate with the super versions: Sf_n = \frac{Sg^n-Sc^n}{Sg-Sc} . After looking at the pattern I arrived at my answer:

Sf_n = Sf_{n-1} + Sf_{n-3} where Sf_0 = 0 , Sf_1 = 1 and Sf_2 = 1
\boldsymbol{Sf} = (0,1,1,1,2,3,4,6,9,13,22,31,44,66,97,\dots)

\displaystyle \lim_{n \to \infty}\frac{{Sf}_{n+1}}{{Sf}_{n}} = Sg \approx 1.4655

The Extended Fibonacci Numbers:
Before I derived the Super numbers from the Super ratio I thought maybe they would be the same as the Fibonacci but with an extra term: Sf_n = Sf_{n-1} + Sf_{n-2} + Sf_{n-3} …it wasn’t. The sequence, which I am calling the Extended Fibonacci numbers (Ef), looked like this:

Ef = (0,1,1,2,4,7,13,68,81,149,274,504,927,1705,3136, \dots)

\displaystyle \lim_{n \to \infty}\frac{{Ef}_{n+1}}{{Ef}_{n}} = Eg \approx 1.8392

I looked back at my scribblings and noticed that the polynomial didn’t just have an extra term on it like I thought, an x was also removed. I whipped up WolfRamAlpha and punched in x^3 - x^2 - x - 1 = 0 (putting the -x back in) and needless to say x \approx 1.83929

Related reading:


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