Some q-congruences related to 3-adic valuations

In 1992, Strauss, Shallit and Zagier proved that for any positive integer a,∑k=03a−1(2kk)≡0(mod32a) and furthermore132a∑k=03a−1(2kk)≡1(mod3). Recently a q-analogue of the first congruence was conjectured by Guo and Zeng. In this paper we prove the conjecture of Guo and Zeng, and also give a q-analogue of the second congruence.