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We prove that $\sum_{k=0}^{q-1}\binom{2k}{k}\equiv q^2\pmod{3q^2}$ if q>1 isa power of 3, as recently conjectured by Z.W. Sun and R. Tauraso. Our moreprecise 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 isindependent of q, for q a power of 3 large enough, and shows how such value canbe efficiently computed.

The sum $\sum_{k=0}^{q-1}\binom{2k}{k}$ for q a power of 3