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In this paper, we study the combinatorial sum $$\sum_{k\equiv r(\mbox{mod}m)}{n\choose k}a^k.$$ By studying this sum, we obtain new congruences forLucas quotients of two infinite families of Lucas sequences. Only for threeLucas sequences, there are such known results.

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