Affiliation:
1. Department of Mathematics University of Haifa 3498838 Haifa Israel
2. Department of Mathematics University of Tennessee Knoxville , TN 37996 U.S.A
Abstract
Abstract
Given k ≥ 2, let a
n
be the sequence defined by the recurrence a
n
= α
1
a
n–1 + … + α
k
a
n–k
for n ≥ k, with initial values a
0 = a
1 = … = a
k–2 = 0 and a
k–1 = 1. We show under a couple of assumptions concerning the constants α
i
that the ratio
a
n
n
a
n
−
1
n
−
1
$\frac{\sqrt[n]{a_n}}{\sqrt[n-1]{a_{n-1}}}$
is strictly decreasing for all n ≥ N, for some N depending on the sequence, and has limit 1. In particular, this holds in the cases when all of the α
i
are unity or when all of the α
i
are zero except for the first and last, which are unity. Furthermore, when k = 3 or k = 4, it is shown that one may take N to be an integer less than 12 in each of these cases.
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