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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 558.10010

**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) = **sum**^{s}_{i = 1} r_{i}a([n/m_{i}]) for n \geq 1, where r_{i} > 0 and the m_{i} are integers \geq 2, the asymptotic behavior of a(n) is determined. Let \tau be the unique solution to **sum**^{s}_{i = 1} r_{i}m_{i}^{-\tau} = 1. When there isan integer d and integers u_{i} such that m_{i} = d^{ui} for all i, a(n)/n^{\tau} oscillates, while in the other case, where no such d and u_{i} 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.

**Reviewer: ** P.Erdös

**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

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