We find the exact results for the reduced entropies using the transfer matrix method, previously applied to find correlations (see [87, 30]). Write the energy 
symmetrically as
Define the 
transfer matrix L by
With
we have
or
In terms of the transfer matrix L the partition function is given by
These forms may be simplified directly, noting that the sums of the products above amount to matrix products, so that we have
and
