  In this article we study $p$-adic properties of sequences of integers (or
$p$-adic integers) that satisfy a linear recurrence with constant coefficients.
For such a sequence, we give an explicit approximate twisted interpolation to
$\mathbb Z_p$. We then use this interpolation for two applications. The first
is that certain subsequences of constant-recursive sequences converge
$p$-adically. The second is that the density of the residues modulo $p^\alpha$
attained by a constant-recursive sequence converges, as $\alpha \to \infty$, to
the Haar measure of a certain subset of $\mathbb Z_p$. To illustrate these
results, we determine some particular limits for the Fibonacci sequence.
