  IMPORTANT NOTE: This paper is much rougher than I'd usually submit, and not
entirely complete, though the main theorems and proofs should not be hard to
follow. Given the ongoing strike at UK Universities it may be some time before
I get to complete it to my satisfaction, and in the meantime people I've shared
the preliminary draft with would like to be able to reference it. Hence I'm
uploading it in its current form, and will update it later.
  The main new result of this paper is to count the number of (n,n+1)-core
partitions with odd parts, answering a question of Zaleski and Zeilberger with
bounty a charitable contribution to the OEIS. Along the way, we prove a general
theorem giving a recurrence for (n,n+1)-core parts whose smallest part and
consecutive part differences are restricted to lie in an arbitrary set M. This
theorem unifies many known results about (n,n+1)-core partitions with
restrictions.
  We end with discussions of extensions of the general theorem that keep track
of the largest part, number of parts, and size of the partition, and about a
few cases where the same methods work on more general simultaneous cores.
