  Bayart and Ruzsa [Ergodic Theory Dynam. Systems 35 (2015)] have recently
shown that every frequently hypercyclic weighted shift on $\ell^p$ is chaotic.
This contrasts with an earlier result of Bayart and Grivaux [Proc. London Math.
Soc. (3) 94 (2007)] who constructed a non-chaotic frequently hypercyclic
weighted shift on $c_0$. We first generalize the Bayart-Ruzsa theorem to all
Banach sequence spaces in which the unit sequences are a boundedly complete
unconditional basis. We then study the relationship between frequent
hypercyclicity and chaos for weighted shifts on Fr\'echet sequence spaces, in
particular on K\"othe sequence spaces, and then on the special class of power
series spaces. We obtain, rather curiously, that every frequently hypercyclic
weighted shift on $H(\mathbb{D})$ is chaotic, while $H(\mathbb{C})$ admits a
non-chaotic frequently hypercyclic weighted shift.
