About definition of recursive sequence Can I define the recursive sequence in the following?


*

*Let $f: \Bbb{N} \to \Bbb{R}$ a sequence and $k \in \Bbb{N}$ and $\forall x \in \Bbb{N}(f_x:=f(x))$ , $f$ is recursive sequence if $$\begin{cases} f_0=a_0 \wedge a_0 \in \Bbb{R}\\ f_1=a_1 \wedge a_1 \in \Bbb{R} \\ . \\. \\f_k=a_k \wedge a_k \in \Bbb{R} \\\forall n \in \Bbb{N}(n \geq k\to f_{(n+1)}=f_{(n)} )
\end{cases} $$


Thanks in advance!

A sequence $f$ is recursive if there is some function $G:\Bbb R^{k+1}\to \Bbb R$ so that
$$
f_{n+1}=G(f_n,f_{n-1},...,f_{n-k})
$$
Then the sequence is determined by providing initial values $f_i=a_i$ for $i=0,1,...,k-1,k$.
So no, your definition is strange and seems to define a sequence that is constant for $n>k$.
What exactly is the question?
