  We study corrections to single tetrahedron based approximations for the
entropy, specific heat and uniform susceptibility of the pyrochlore lattice
Ising antiferromagnet, by a Numerical Linked Cluster (NLC) expansion. In a
tetrahedron based NLC, the first order gives the Pauling residual entropy of
${1\over 2}\log{3\over 2}\approx 0.20273$. A 16-th order NLC calculation
changes the residual entropy to 0.205507 a correction of 1.37 percent over the
Pauling value. At high temperatures, the accuracy of the calculations is
verified by a high temperature series expansion. We find the corrections to the
single tetrahedron approximations to be at most a few percent for all the
thermodynamic properties.
