fibonacci like

[bo198214]

then its not a recursion, i.e. you dont have a unique solution for natural number arguments.
I dont think the usual methods apply.

i didnt claim unique solutions.

...

why isnt it a recursion ???

A recursion needs to be determined on the natural numbers (or in other words unique), i.e. you can trace back every f(n) to some initial given f(0) or f(1) or so.
Like in the fibonacci numbers
f(n)=f(n-1)+f(n-2),f(0)=0,f(1)=1

Or even put it differently: when you write a computer program with the defining relation it must terminate.
Yours is not a recursion just some functional equation.