[Paper Review] A family of meta-Fibonacci sequences defined by variable order recursions

Nathaniel D. Emerson|ArXiv.org|Aug 26, 2005

Advanced Mathematical Theories and Applications10 references3 citations

TL;DR

This paper introduces a new family of meta-Fibonacci sequences—variable-r-bonacci sequences—where the recursion order r(n) varies with n, and each term b(n) is the sum of the previous r(n) terms. It establishes tight closed-form upper and lower bounds on b(n) using two key functions, λ(n) and μ(n,s), proving that the growth rate of b(n) is bounded by these terms, enabling precise estimation of sequence behavior across diverse growth regimes including exponential, linear, and logarithmic.

ABSTRACT

We define a family of meta-Fibonacci sequences where the order of the of recursion at stage n is a variable r(n), and the n^{th} term of a sequence is the sum of the previous r(n) terms. For the terms of any such sequence, we give upper and lower bounds which depend only on r(n).

Motivation & Objective

• To define and analyze a new family of meta-Fibonacci sequences where the recursion order r(n) varies with n.
• To address the lack of general bounds for sequences with variable recursion orders, especially when r(n) is sublinear.
• To provide closed-form estimates for b(n) that depend only on the sequence of recursion orders r(1), ..., r(n).
• To characterize the full spectrum of possible growth rates—exponential, linear, logarithmic—within this family.
• To generalize the framework to allow negative indices and arbitrary initial conditions, extending applicability beyond standard sequences.

Proposed method

• Define variable-r-bonacci sequences via the recurrence b(n) = Σ_{k=1}^{r(n)} b(n-k) for n > 1, with b(0) = 1 and r(n) sublinear.
• Introduce two auxiliary functions: λ(n) = 1 + (r(n) - 1)/r(n-1) and μ(n,s) = 2 + [r(n) - r(n-1) - 1] × ∏_{k=n-s}^{n-1} 1/r(k), which capture growth dynamics.
• Establish the Main Theorem: the ratio b(n)/b(n-1) is bounded between min{λ(n), μ(n,r(n)-1)} and max{λ(n), μ(n,r(n-1))}.
• Derive product-based bounds on b(n) by telescoping the ratio estimates from n=2 to n.
• Generalize the sequence to all integers by extending r(n) to Z and defining β(n) recursively backward, allowing arbitrary initial conditions.
• Prove that for r(n) with finite M_r = sup(r(n) - n), the extended sequences form a real vector space of dimension M_r.

Experimental results

Research questions

• RQ1 How does the growth rate of a meta-Fibonacci sequence behave when the recursion order r(n) varies with n rather than being fixed?
• RQ2 Can tight, closed-form upper and lower bounds be derived for b(n) that depend only on the sequence of recursion orders r(1), ..., r(n)?
• RQ3 What range of growth behaviors—exponential, linear, logarithmic—is possible in variable-order meta-Fibonacci sequences?
• RQ4 Can the sequence be extended to negative indices while preserving the recurrence and allowing arbitrary initial conditions?
• RQ5 What is the dimension of the solution space for such sequences when generalized to all integers?

Key findings

• The ratio b(n)/b(n-1) is bounded between min{λ(n), μ(n,r(n)-1)} and max{λ(n), μ(n,r(n-1))}, providing precise control over growth at each step.
• The sequence b(n) grows at most exponentially, and if r(n) > 1 for all sufficiently large n, it grows at least exponentially.
• Linear growth is possible, but only if r(n) = 1 infinitely often; in such cases, the asymptotic density L = lim b(n)/n must be zero.
• Logarithmic growth, such as Θ(log₂ n) or slower, is achievable by setting r(n) = 2 only at sparse points like n = 2^{2^k}.
• The extended variable-r-bonacci sequences form a real vector space of dimension M_r = sup(r(n) - n), generalizing the original family.
• For sublinear r(n), M_r = 1, so the solution space is one-dimensional, parameterized by β(-1).