A Family of Meta-Fibonacci Sequences Defined by Variable-Order Recursions

We define a family of meta-Fibonacci sequences. For each sequence in the family, the order of the of the defining recursion at the nth stage is a variable r(n), and the nth 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.