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:

where and

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:

**The Lucas Numbers:**

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

where and

The Lucas Numbers relate directly to the Fibonaci Numbers, for example: and even though almost all the numbers are different, they both converge to the same ratio:

**The Pell Numbers:**

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

where and

The Pell numbers related to which can be seen when the series converges to the Silver Ratio :

**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:

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:

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 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 and has the value of

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: . After looking at the pattern I arrived at my answer:

where , and

**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: …it wasn’t. The sequence, which I am calling the Extended Fibonacci numbers (Ef), looked like this:

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 (putting the -x back in) and needless to say

**Related reading:**

- Fibonacci numbers and the Golden ratio – thecodeaddict
- http://en.wikipedia.org/wiki/Fibonacci_number
- http://mathworld.wolfram.com/LucasNumber.html
- http://mathworld.wolfram.com/PellNumber.html
- http://en.wikipedia.org/wiki/Look-and-say_sequence
- 50 Mathematical Ideas You Really Need to Know – Tony Crilly (Book)

This is why I studied history…

lol, do you have a favourite subject in history?