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On Families of Nonlinear Recurrences Related to Digits
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Th. Stoll

Faculty of Mathematics

University of Vienna

Nordbergstraße 15

1090 Vienna

Austria

and

Institute of Discrete Mathematics and Geometry

Wiedner Hauptstraße 8-10

1040 Vienna

Austria

**Abstract:**
Consider the sequence of positive integers
defined by and
.
Graham and Pollak discovered the unexpected fact that
is just the -th digit in the binary
expansion of . Fix
. In this note, we
first give two infinite families of similar nonlinear recurrences
such that
indicates the -th binary digit
of . Moreover, for all integral , we establish a
recurrence such that
denotes the -th digit
of in the -ary digital expansion.

(Concerned with sequences
A001521
A091522 and
A091523
.)

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Received April 1 2005;
revised version received May 12 2005.
Published in *Journal of Integer Sequences* May 24 2005.

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