  Inspired by possibility that the Schwarzschild black hole may not be the
unique spherically symmetric vacuum solution to generalisations of general
relativity, we consider black holes in pure fourth order higher derivative
gravity treated as an effective theory. Such solutions may be of interest in
addressing the issue of higher derivative hair or during the later stages of
black hole evaporation. Non-Schwarzschild solutions have been studied but we
have put earlier results on a firmer footing by finding a systematic asymptotic
expansion for the black holes and matching them with known numerical solutions
obtained by integrating out from the near horizon region. These asymptotic
expansions can be cast in the form of trans-series expansions which we
conjecture will be a generic feature of non-Schwarzschild higher derivative
black holes. Excitingly we find a new branch of solutions with lower free
energy than the Schwarzschild solution, but as found in earlier work, solutions
only seem to exist for black holes with large curvatures meaning that one
should not generically neglect even higher derivative corrections. This
suggests that one effectively recovers the non-hair theorems in this context.
