  We prove that $\sum_{k=0}^{q-1}\binom{2k}{k}\equiv q^2\pmod{3q^2}$ if q>1 is
a power of 3, as recently conjectured by Z.W. Sun and R. Tauraso. Our more
precise result actually implies that the value of
$(1/q^2)\sum_{k=0}^{q-1}\binom{2k}{k}$ modulo a fixed arbitrary power of 3 is
independent of q, for q a power of 3 large enough, and shows how such value can
be efficiently computed.
