## TR #408: Convergence of generalized Fibonacci sequences

**
Matthew Brand
** November 1996
Submitted to:

The Fibonacci Quarterly

We generalize the Fibonacci sequence *F*_{i}=
F_{i-1}+F_{i-2} in two ways: arbitrary generating
sum-rules (*F*_{i}=k_{1}F_{i-1}+...) and
exponential attenuation (*G*_{i}=F_{i}/c^{i}).
We show under what conditions such series will converge and identify series
that converge to interesting constants, including, of course, *phi*.
The generalizations allow us to model varying maturation, reproduction, and
mortality rates for populations, and identify steady-state and explosive
growth regimes. An added curiousity is that these series also give rise to
a set of fractions (*1/(10*^{2k}-10^{k}-1), e.g.,
*1/9899*), whose decimal expansion includes an embedded Fibonacci
series.