Everyone is familiar with the Fibonacci Sequence, [0] 1 1 2 3 5 8 ...
and many of it's interesting properties. For example, as the sequence continues, the ratio of $\frac{F_n}{F_{n-1}}$ converges to $\tau=\frac{1+\sqrt{5}}{2}$, a ratio which can be used to describe a number of numerical relationships in nature.
From some quick Wikipedia browsing, we can find a number of Generalizations of Fibonacci numbers. For one abstract generalization, we can define Fibonacci-like sequences as follows:
An $n$-order Fibonacci-like sequence is generated by $F_n=\sum_{i=1}^{n}F_{n-i}$ with $n$ initial terms.
Thus, the Fibonacci sequence is such a sequence with $n=2$ and $F_0=0$ and $F_1=1$
Using this basic generalization, we have Lucas Numbers, where $n=2$ and $F_0=2$ and $F_1=1$, whose consecutive-number ratio also converges to the golden ratio. There are also Tribonacci, Tetranacci, and n-nacci numbers, which follow this generalization for 3, 4, and n numbers, and pad the initializing values with 0s, e.g. $F_0=0$ ... $F_{n-2}=0$, $F_{n-1}=1$.
So, my question is, are there any important properties of these sequences that are worth learning? Do these sequences have real-world applications or reflections like the original Fibonacci sequence does? What can be learned from these?