Pick four integers $a,b,c$ and $d$. Then we get a corresponding sequence given by $$t_{n+2} = at_{n+1} +bt_n, \; t_1 = c, \;t_2 = d.$$
From what I can tell, we seem to get an especially rich theory when we choose $a=1,b=1,c=1,d=1$, thereby obtaining the Fibonacci sequence. Just take a look at the relevant wikipedia page; it's simply huge, and full of interesting-looking identities and connections.
Question. Why is this? What is about these four numbers that gives such a rich theory for the corresponding sequence?
A good answer should either:
Explain that most of the results about the Fibonacci sequence have analogs that work for any $a,b,c$ and $d$ satisfying some weak conditions, so really the Fibonacci sequence isn't that special, or:
Specify a very strong constraint on the relationship between $a,b,c$ and $d$ and explain why this constraint makes this particular sequence and the (few) others like it to have a very rich theory.
In calculus and differential equations, the most important functions are those which are their own derivatives: $e^x, \sin x, \cosh x, $ etc. The most fundamental is $e^x$ and the others can be expressed in terms of it. It satisfies the equation $y' = y.$
The Fibonacci sequence is its own difference sequence. (Write down the sequence and then write the differences between each pair of successive terms, and you'll get another copy of the Fibonacci sequence.) So it satisfies the equation $F = \Delta F.$ So it's not a surprise that solutions to difference equations can be expressed in terms of the Fibonacci sequence. Your recursive definition involving $a,b, c$ and $d$ is really a difference equation, as is the definition of the Tribonacci sequence and other more general things.