Asymptotics of Lagged Fibonacci Sequences

TL;DR

This paper analyzes the asymptotic behavior of lagged Fibonacci sequences, deriving a limit relation, discussing convergence issues, and connecting classical results on their asymptotics.

Contribution

It provides a new asymptotic limit for lagged Fibonacci sequences and discusses convergence and classical connections.

Findings

Limit: $a(kn)/a(n) o (k o ext{large})$ with a specific asymptotic relation.

Demonstrates slow numerical convergence and methods to address it.

Connects results of de Bruijn and Mahler on sequence asymptotics.

Abstract

Consider "lagged" Fibonacci sequences $a(n) = a(n-1)+a(\lfloor n/k\rfloor)$ for $k > 1$. We show that $\lim_{n\to\infty} a(kn)/a(n)\cdot\ln n/n = k\ln k$ and we demonstrate the slow numerical convergence to this limit and how to deal with this slow convergence. We also discuss the connection between two classical results of N.G. de Bruijn and K. Mahler on the asymptotics of $a(n)$.