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A Family of Meta-Fibonacci Sequences Defined by Variable-Order Recursions
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Nathaniel D. Emerson

Department of Mathematics

California State University, Channel Islands

One University Drive

Camarillo, California 93012-8599

USA

**Abstract:**
We define a family of meta-Fibonacci sequences. For each sequence in
the family, the order of the of the defining
recursion at the
*n*^{th}
stage is a variable *r(n)*, and the
*n*^{th}
term is the sum of the
previous *r(n)* terms. Given a sequence of real numbers that satisfies
some conditions on growth, there is a meta-Fibonacci sequence in the
family that grows at the same rate as the given sequence. In
particular, the growth rate of these sequences can be exponential,
polynomial, or logarithmic. However, the possible asymptotic limits of
such a sequence are restricted to a class of exponential functions. We
give upper and lower bounds for the terms of any such sequence, which
depend only on *r(n)*. The Narayana-Zidek-Capell sequence is a member
of this family. We show that it converges asymptotically.

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(Concerned with sequences
A000045
A000073
A002083
A004001
A005185
A006949
and
A092921
.)

Received September 13 2005;
revised version received March 17 2006.
Published in *Journal of Integer Sequences* March 17 2006.

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