It is well-known that the weighted backshift operator $B_{\lambda}:\ell^p \rightarrow \ell^p$ is hypercyclic (with $\lambda>1$); that is, there exists a dense set of sequences $X\subseteq \ell^p$ for which $$ \overline{\left\{B^n(x)\right\}_{n \in \mathbb{N}} } = \ell^p \qquad (\forall x \in X). $$
Is there a known, concrete example of such an $x\in X$?
Probably not an example you are looking for. Rolewicz's proof is essentially an application of Baire's theorem and you can make this in a certain sense construtive: Choose a dense countable set $\{x^k: k\in \mathbb N\}$ of finite sequences (i.e., terminating with zeros) in $\ell^p$, align $y^k=(x^k_1,\lambda x^k_2, \lambda^2 x^k_3,\ldots)$ in a single sequence separated by many zeros and mutiply the $n$-th term of this sequence by $1/\lambda^{n-1}$. Choosing the blocks of separating zeros long enough will make this sequence an element of $\ell^p$. Given $k$ you apply $\lambda B$ many times to get the sequence $x^k$ in front followed by many zeros and a rest with very small $p$-norm.
The missing details are very ugly. The only advantage compared to the much more elegant use of Baire's theorem is that you can apply such a kind of construction to other locally convex sequence spaces as, e.g., in Bonet, J.; Frerick, L.; Peris, A.; Wengenroth, J. Transitive and hypercyclic operators on locally convex spaces, Bull. Lond. Math. Soc. 37, No. 2, 254-264 (2005).
