  We consider two families of algebraic varieties $Y_n$ indexed by natural
numbers $n$: the configuration space of unordered $n$-tuples of distinct points
on $\mathbb{C}$, and the space of unordered $n$-tuples of linearly independent
lines in $\mathbb{C}^n$. Let $W_n$ be any sequence of virtual
$S_n$-representations given by a character polynomial, we compute $H^i(Y_n;
W_n)$ for all $i$ and all $n$ in terms of double generating functions. One
consequence of the computation is a new recurrence phenomenon: the stable
twisted Betti numbers $\lim_{n\to\infty}\dim H^i(Y_n; W_n)$ are linearly
recurrent in $i$. Our method is to compute twisted point-counts on the
$F_q$-points of certain algebraic varieties, and then pass through the
Grothendieck-Lefschetz fixed point formula to prove results in topology. We
also generalize a result of Church-Ellenberg-Farb about the configuration
spaces of the affine line to those of a general smooth variety.
