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Complete Sequences of Weighted ***r*-Generalized Fibonacci Powers

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Passawan Noppakaew and Thanakorn Prinyasart

Department of Mathematics

Faculty of Science

Silpakorn University

Nakhon Pathom 73000

Thailand

**Abstract:**

In this paper, we study the least number *L* ∈ **N** that makes
the *L*-fold of a weighted *r*-generalized Fibonacci power
complete. We can establish an upper bound and a lower bound for *L*,
depending on the first *r* terms and the limit of the consecutive
ratios. We also give the explicit value of *L* in some special
cases. In the end, we study a particular minimal complete sequence
called a generalized distinguished sequence and show that there exists
a generalized distinguished sequence of a weighted *r*-generalized
Fibonacci *m*th power for all large *m* ∈ **N**.

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(Concerned with sequences
A000032
A000045
A007598.)

Received September 13 2020; revised versions received September 14 2020; February 10 2021; February 26 2021.
Published in *Journal of Integer Sequences*,
April 25 2021.

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