  We report measurements of the anisotropy of the spin echo decay for the inner
layer Cu site of the triple layer cuprate, Hg_0.8Re_0.2Ba_2Ca_2Cu_3O_8 (T_c=126
K) in the pseudogap T regime below T_pg ~ 170 K and the corresponding analysis
for their interpretation. As the field alignment is varied, the shape of the
decay curve changes from Gaussian (H_0 \parallel c) to single exponential (H_0
\perp c). The latter characterizes the decay caused by the fluctuations of
adjacent Cu nuclear spins caused by their interactions with electron spins. The
angular dependence of the second moment (T_{2M}^{-2} \equiv <\Delta \omega^2>)
deduced from the decay curves indicates that T_{2M}^{-2} for H_0 \parallel c,
which is identical to T_{2G}^{-2} (T_{2G} is the Gaussian component), is
substantially enhanced, as seen in the pseudogap regime of the bilayer systems.
Comparison of T_{2M}^{-2} between H_0 \parallel c and H_0 \perp c indicates
that this enhancement is caused by electron spin correlations between the inner
and the outer CuO_2 layers. These results provide the answer to the
long-standing controversy regarding the opposite T dependences of (T_1T)^{-1}
and T_{2G}^{-2} in the pseudogap regime of bi- and trilayer systems.
