It is well-known that the backshift operator $B:\ell^p \rightarrow \ell^p$ is hypercyclic; 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 $x\in X$?

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 $x\in X$?