A sum of Fibonacci numbers

Using Wolfram for these sums, we get approximate values \begin{align*} C \approx 0.86082, \qquad B \approx -0.11649, \qquad A \approx -0.07897 \end{align*} From there, you can collect coefficients of $F_r$ to prove your identity. A more concrete way to derive exact expressions is through generating functions. Define \begin{align*} g(x) = \sum_{n=0}^{\infty}\frac{(-1)^n}{\binom{2n}{n}(2n+1)}x^n \end{align*} which you can show equals \begin{align*} g(x) = 4 \frac{\sinh^{-1}(\sqrt{x}/2)}{\sqrt{x(x+4)}} \end{align*} If we define $L = x \frac{d}{dx}$ as the operator which first takes the derivative w.r.t $x$, then multiplies by $x$, we get that \begin{align*} C = (L^0 g)(1), \qquad B = (L^1 g)(1), \qquad A = (L^2 g)(1) \end{align*} or \begin{align*} C = 4 \frac{\sinh^{-1}(1/2)}{\sqrt{5}}, \qquad B = \frac{2}{5} - \frac{12 \sinh^{-1}(1/2)}{5\sqrt{5}}, \qquad A = \frac{4}{125}(7\sqrt{5}\sinh^{-1}(1/2) - 10) \end{align*}