Consider the real vector space of infinite sequences of real numbers
\[
S=\left\{\left(a_{0}, a_{1}, a_{2}, \ldots\right) \mid a_{k} \in \mathbb{R}, k=0,1,2, \ldots\right\} .
\]
Let $W$ be the subspace of $S$ consisting of all sequences $\left(a_{0}, a_{1}, a_{2}, \ldots\right)$ which satisfy the relation
\[
a_{k+2}=2 a_{k+1}+a_{k}, \quad k=0,1,2, \ldots
\]
What is the dimension of $W$ ?
- $1$
- $2$
- $3$
- $\infty$