Series and sums involving the floor function

Let $(a_n{)}_{n\ge 0}$ be an arbitrary sequence and $(a_{\lfloor n\mathrm{/}k\rfloor }{)}_{n\ge 0}$ its dual floor sequence. We study infinite series and finite generalized binomial sums involving $(a_{\lfloor n\mathrm{/}k\rfloor }{)}_{n\ge 0}$. As applications we prove a range of new closed form expressions for Fibonacci (Lucas) series and binomial sum identities as particular cases.