2023/09/22 by Carlo Sanna, Sanna, Carlo
Computer Science · Mathematics · #05A17 #11B39 (Primary) 05A16 #11P99 #68Q45 (Secondary) #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #FOS: Computer and information sciences #FOS: Mathematics #Formal Languages and Automata Theory (cs.FL) #Number Theory (math.NT) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2309.12724
openalex publication_date 2023/09/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For every nonnegative integer n, let rF(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 SF(p)(N) := ∑n = 0N - 1 (rF(n))p . Chow, Jones, and Slattery determined the order of growth of SF(p)(N) for p ∈ \1,2\. We prove that, for all positive integers p, there exists a real number λp > 1 such that S(p)F(N) \asympp N(log λp) / log φ as N → +∞, where φ:= (1 + √(5))/2 is the golden ratio. Furthermore, we show that limp → +∞ λp1/p = φ1/2 . Our proofs employ automata theory and a result on the generalized spectral radius due to Blondel and Nesterov.