  We identify the weights $wt(f_n)$ of a family $\{f_n\}$ of rotation symmetric
Boolean functions with the cardinalities of the sets of $n$-periodic points of
a finite-type shift, recovering the second author's result that said weights
satisfy a linear recurrence. Similarly, the weights of idempotent functions
$f_n$ defined on finite fields can be recovered as the cardinalities of curves
over those fields and hence satisfy a linear recurrence as a consequence of the
rationality of curves' zeta functions. Weil's Riemann hypothesis for curves
then provides additional information about $wt(f_n)$. We apply our results to
the case of quadratic functions and considerably extend the results in an
earlier paper of ours.
