A family of meta-Fibonacci sequences defined by variable order recursions

TL;DR

This paper introduces a new family of meta-Fibonacci sequences with variable recursion order, providing bounds based solely on the recursion order function, expanding understanding of such recursive sequences.

Contribution

It defines a novel class of meta-Fibonacci sequences with variable order and derives bounds that depend only on the order function r(n).

Findings

Established upper and lower bounds for the sequences.

Sequences are characterized by variable recursion order r(n).

Results depend solely on the function r(n).

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).