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