Let us: $\{f_k\}$ Fibonacci sequence; $\{u_k\}$ binary sequence i.e. $u_k=$ 0 or 1; $\rho$ real positive number. Is there a binary sequence $v_k$ such that:
$$\sum_{k=2}^{n+1}\frac{u_{k-1} f_{k-1}}{\rho^{k+1}}+\sum_{k=1}^{n+1}\frac{u_{k} f_{k}}{\rho^{k+1}}=\sum_{k=1}^{n+1}\frac{v_{k+1} f_{k+1}}{\rho^{k+1}}$$ ?
Not sure if I'm missing something in the question but it would seem that there are sequences $u_k$ for which a suitable sequence $v_k$ does not exist. Take for example $u_1=1$, $u_2=0$. With $f_0=0$, $f_1=1$, $f_2=1$, $f_3=2$, and any $\rho$, the equations for $n=0$ and $n=1$ are $$\frac{1}{\rho^2}=\frac{v_2}{\rho^2}\quad\hbox{and}\quad \frac{1}{\rho^3}+\frac{1}{\rho^2}=\frac{v_2}{\rho^2}+\frac{2v_3}{\rho^3}\ ;$$ and this gives $2v_3=1$, which is impossible if all the $v_k$ have to be $0$ or $1$.