  In this paper, we use the dimer method to obtain the free energy of Ising
models consisting of repeated horizontal strips of width $m$ connected by
sequences of vertical strings of length $n$ mutually separated by distance $N$,
with $N$ arbitrary, to investigate the effects of connectivity and proximity on
the specific heat. The decoration method is used to transform the strings of
$n+1$ spins interacting with their nearest neighbors with coupling $J$ into a
pair with coupling $\bar J$ between the two spins. The free energy per site is
given as a single integral and some results for critical temperatures are
derived.
