A strange property of lattices with an even number of sites
  By examining the behaviour of the "SLAC" lattice derivative operators, it is found that lattices with an even number of sites have a somewhat strange self-consistency requirement for extra structure in the spatial derivative operator, which is not needed by lattices having an odd number of sites, and which is not at all obvious from a first-principles derivation. The general implications of this extra required structure are not, as yet, completely clear. 
John Costella