A note on the power sums of the number of Fibonacci partitions
Carlo Sanna
TL;DR
The paper proves that for each p ≥ 1, the p-th power sum S^{(p)}_F(N) of Fibonacci partitions satisfies S^{(p)}_F(N) ∼_p N^{(\\log \\lambda_p)/\\log \\varphi} with λ_p > 1, and establishes the asymptotic limit lim_{p→∞} λ_p^{1/p} = φ^{1/2}. The authors derive this via automata-theoretic encoding of Fibonacci representations, constructing a_p whose PF eigenvalue λ_p governs the growth, and they connect λ_p to the generalized spectral radius of matrices arising from Berstel’s automaton through Kronecker powers. Moreover, λ_p is shown to be the largest root of an effectively computable monic integer polynomial, making λ_p an algebraic integer and enabling explicit computation. The methods, including the Berstel automaton and Blondel–Nesterov’s radius results, extend to other linear recurrences, highlighting a broad applicability to partition-related growth problems.
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 $λ_p > 1$ such that \begin{equation*} S^{(p)}_F(N) \asymp_p N^{(\log λ_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} λ_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.