DISCLAIMER: I do not deal with in-depth mathematics on a daily basis as some of you may, so please pardon my ignorance or lack of coherence on this topic.
QUESTION: What is the significance of starting the Fibonacci sequence with $0,1$ ?
For instance, if I picked any two random integers, say 2 and 7, to start a sequence would I actually be creating some multiple or derivation of the Fibonacci sequence?
Is there a general mathematical explanation for the relationship between any sequence represented by $a[0] = x, a[1] = y, a[n] = a[n-1] + a[n-2]$ and the Fibonacci sequence?
Or, back to my example sequence, is there a general mathematical relationship between:
$2,7,9,16,25,41,66,107,173,280…$
and
$0,1,1,2,3,5,8,13,21,34…$
Perhaps the Golden Ratio explains it somehow? Any help would be appreciated.
Best Answer
Yes, such sequences are closely related, and the relationship does involve the golden ratio.
Let $\varphi=\frac12(1+\sqrt5)$ and $\widehat\varphi=\frac12(1-\sqrt5)$; $\varphi$ is of course the golden ratio, and $\widehat\varphi$ is its negative reciprocal. Let $a_0$ and $a_1$ be arbitrary, and define a Fibonacci-like sequence by the recurrence $a_n=a_{n-1}+a_{n-2}$ for $n\ge 2$. Then there are constants $\alpha$ and $\beta$ such that
$$a_n=\alpha\varphi^n+\beta\widehat\varphi^n\tag{1}$$
for each $n\ge 0$. Indeed, you can find them by substituting $n=0$ and $n=1$ into $(1)$ and solving the system
$$\left\{\begin{align*} a_0&=\alpha+\beta\\ a_1&=\alpha\varphi+\beta\widehat\varphi \end{align*}\right.$$
for $\alpha$ and $\beta$. In the case of the Fibonacci numbers themselves, $\alpha=\frac1{\sqrt5}$ and $\beta=-\frac1{\sqrt5}$; in the case of the Lucas numbers $L_n$, for which the initial values are $L_0=2$ and $L_1=1$, $\alpha=\beta=1$.