  We construct multipole moments for stationary, asymptotically flat, spacetime
solutions to higher-order curvature theories of gravity. The moments are
defined using $3+1$ techniques involving timelike Killing vector constructions
as in the classic papers by Geroch and Hansen. Using the fact that the
Kerr-Newman metric is a vacuum solution to a particular class of $f(R)$
theories of gravity, we compute all its moments, and find that they admit
recurrence relations similar to those for the Kerr solution in general
relativity. It has been proposed previously that modelling the measured
frequencies of quasi-periodic oscillations from galactic microquasars enables
experimental tests of the no-hair theorem. We explore the possibility that,
even if the no-hair relation is found to break down in the context of general
relativity, there may be an $f(R)$ counterpart that is preserved. We apply the
results to the microquasars GRS $1915$+$105$ and GRO J$1655$-$40$ using the
diskoseismology and kinematic resonance models, and constrain the spins and
`charges' [which are not really electric charges in the $f(R)$ context] of
their black holes.
