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 λ_p) /\!\log φ} \end{equation*} as $N\to +\infty$, where $\phi :=(1+\sqrt{5})/2$ is the golden ratio. Furthermore, we show that \begin{equation*} \lim_{p \to +\infty} λ_p^{1/p} = φ^{1/2} . \end{equation*} Our proofs employ automata theory and a result on the generalized spectral radius due to Blondel and Nesterov.