Publications of (and about) Paul Erdös
Autor: Erdös, Paul; Hildebrand, A.; Odlyzko, Andrew M.; Pudaite, P.; Reznick, B.
Title: The asymptotic behavior of a family of sequences. (In English)
Source: Pac. J. Math. 126, No.2, 227-241 (1987).
Review: A class of sequences defined by nonlinear recurrences involving the greatest integer function [.] is studied, a typical member of the class being a(0) = 1, a(n) = a([n/2])+a([n/3])+a([n/6]) for n \geq 1. For this sequence, it is shown that lim a(n)/n as n > oo exists and equals 12/(log 432). More generally, for any sequence defined by a(0) = 1, a(n) = sumsi = 1 ria([n/mi]) for n \geq 1, where ri > 0 and the mi are integers \geq 2, the asymptotic behavior of a(n) is determined. Let \tau be the unique solution to sumsi = 1 rimi-\tau = 1. When there isan integer d and integers ui such that mi = dui for all i, a(n)/n\tau oscillates, while in the other case, where no such d and ui exist, the limit of a(n)/n\tau exists and is explicitly computed. Results on the speed of convergence to the limit are also obtained.
Classif.: * 11B37 Recurrences
11A25 Arithmetic functions, etc.
11B99 Sequences and sets
Keywords: nonlinear recurrences; greatest integer function; asymptotic behaviour; speed of convergence; limit; renewal theory; square functional equation
© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag