Fibonacci zeta in terms of a sum of binomial coefficients
From specialfunctionswiki
Theorem
The following formula holds: $$\left(\phi + \dfrac{1}{\phi} \right)F(z)=-\displaystyle\sum_{j=0}^{\infty} (-1)^j {{-z} \choose j} \dfrac{1}{\phi^{z+2j}+1} ,$$ where $\phi$ denotes the golden ratio, $F(z)$ denotes the Fibonacci zeta function, and ${{-z} \choose j}$ denotes a binomial coefficient.
