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

Abstract: For every nonnegative integer $n$, let $r_F(n)$ be the number of ways to write $n$ as a sum of Fibonacci numbers, where the order of the summands does not 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 $N \to +\infty$, where $\varphi := (1 + \sqrt{5})/2$ is the golden ratio. Furthermore, we show that \begin{equation*} \lim_{p \to +\infty} \lambda_p^{1/p} = \varphi^{1/2}. \end{equation*} Our proofs employ automata theory and a result on the generalized spectral radius due to Blondel and Nesterov.