  We present a variational Monte Carlo (VMC) formulation for the universal
equations of state at the Van der Waals length scale [B. Gao, J. Phys. B
\textbf{37}, L227 (2004)] for $N$ Bose atoms in a trap. The theory illustrates
both how such equations of state can be computed exactly, and the existence and
the importance of long-range atom-atom correlation under strong confinement.
Explicit numerical results are presented for N=3 and 5, and used to provide a
quantitative understanding of the shape-dependent confinement correction that
is important for few atoms under strong confinement.
