Is there a notion of Lucas/Fibonacci sequences beyond adding two numbers at a time, if so where could I read more about that?

Question

So I recently learned about lucas/fibbonaci sequences EG; 1,1,2,3,5,7 or 3,4,7,11,18... A fact I learned about these sequence is that the limit of the quotient of the terms approaches the golden ratio (1.61803398874989484820). What I found independently (somewhat) was that this ratio could also be understood as the root of the polynomial $x^2-x-1$(this can be derived from the geometric definition of the golden ratio).

In trying to understand the golden ratio further, I extended the notion first by finding a new version of the fibbonaci sequence starting with three terms. 1,1,1:3,5,9,17,31,57,105... The ratios of this sequence converge to roughly 1.83, which is also one of the real roots of the polynomial $x^3-x^2-x-1$. At this point, I'm pretty sure that's an obvious pattern.

After I found the ratio for the 3-fibbonaci/lucas sequence I began to wonder what the ratio of an inifnite version of this would converge to (if at all). So I wrote out a general notation for a sequence where you add up n terms starting with (x0,x1,x2,,,,,,xn). The first term of the sequence is the summation from k=0 to n or "X sub K." The second term is the earlier term plus the summation from k=1 to n. As you write out the terms, you can show that the ratio of the terms converges to 2.

Lastly, does this imply that 2 is the root of the polynomial $x^n-x^{n-1}....-x-1?$

Answer 1

Nice work!

You have discovered linear recurrence relations. The place to start reading is wikipedia: https://en.wikipedia.org/wiki/Recurrence_relation

The infinite recurrence you specify does have growth rate $2$ because the $n$th term is $2^{n-1}$ but $2$ is not a root of the polynomials you specify since each expresses just a part of the recursion. For that, see https://en.wikipedia.org/wiki/Composition_(combinatorics)

Answer 2

To answer your actual question independently of what you wrote following it, I think that what you are interested in is the Binet equation (see, for example, Fibonacci Number). The closed form solution for the Fibonacci numbers is given by

$$F_n=\frac{\varphi^n-\psi^n}{\varphi-\psi}$$

and that for the Lucas numbers is given by

$$L_n=\varphi^n+\psi^n$$

where $\varphi$ is the golden ratio and $\psi=1-\varphi$ is its complement.