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In [16], we obtained some congruences for Lucas quotients of two infinitefamilies of Lucas sequences by studying the combinatorial sum $$\sum_{k\equivr(\mbox{mod}m)}{n\choose k}a^k.$$ In this paper, we show that the sum can beexpressed in terms of some recurrent sequences with orders not exceeding$\varphi{(m)}$ and give some new congruences.

Combinatorial Sums $\sum_{k\equiv r(\mbox{mod} m)}{n\choose k}a^k$ and Lucas Quotients (II)