  Chan and Seceleanu have shown that if a weighted shift operator on
$\ell^p(\mathbb{Z})$, $1\leq p<\infty$, admits an orbit with a non-zero limit
point then it is hypercyclic. We present a new proof of this result that allows
to extend it to very general sequence spaces. In a similar vein we show that,
in many sequence spaces, a weighted shift with a non-zero weakly sequentially
recurrent vector has a dense set of such vectors; but an example on
$c_0(\mathbb{Z})$ shows that such an operator is not necessarily hypercyclic.
On the other hand, we obtain that weakly sequentially hypercyclic weighted
shifts are hypercyclic. Chan and Seceleanu have moreover shown that if an
adjoint multiplication operator on a Bergman space admits an orbit with a
non-zero limit point then it is hypercyclic. We extend this result to very
general spaces of analytic functions, including the Hardy spaces.
