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For every nonnegative integer $n$, let $r_F(n)$ be the number of ways towrite $n$ as a sum of Fibonacci numbers, where the order of the summands doesnot matter. Moreover, for all positive integers $p$ and $N$, let\begin{equation*} S_{F}^{(p)}(N) := \sum_{n = 0}^{N - 1} \big(r_F(n)\big)^p .\end{equation*} Chow, Jones, and Slattery determined the order of growth of$S_{F}^{(p)}(N)$ for $p \in \{1,2\}$. We prove that, for all positive integers$p$, there exists a real number $\lambda_p > 1$ such that \begin{equation*}S^{(p)}_F(N) \asymp_p N^{(\log \lambda_p) /\!\log \varphi} \end{equation*} as$N \to +\infty$. Furthermore, we show that egin{equation*} \lim_{p \to +\infty}\lambda_p^{1/p} = \varphi^{1/2} , \end{equation*} where $\varphi := (1 +\sqrt{5})/2$ is the golden ratio. Our proofs employ automata theory and aresult on the generalized spectral radius due to Blondel and Nesterov.

A note on the power sums of the number of Fibonacci partitions