TR #408: Convergence of generalized Fibonacci sequences

Matthew Brand November 1996

Submitted to:

The Fibonacci Quarterly


We generalize the Fibonacci sequence Fi= Fi-1+Fi-2 in two ways: arbitrary generating sum-rules (Fi=k1Fi-1+...) and exponential attenuation (Gi=Fi/ci). 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/(102k-10k-1), e.g., 1/9899), whose decimal expansion includes an embedded Fibonacci series.