On variants of Conway and Conolly's Meta-Fibonacci recursions

TL;DR

This paper investigates variants of Conway and Conolly's meta-Fibonacci recursions, proving conditions for their well-definedness, monotonicity, and establishing connections to generalized Fibonacci-like sequences.

Contribution

It introduces new variants of meta-Fibonacci recursions, proves their well-definedness and monotonicity, and links specific cases to generalized Fibonacci sequences.

Findings

Sequences are well-defined and monotonic under certain initial conditions.

Sequences' forward differences are only 0 or 1.

Certain recursion solutions relate to generalized Fibonacci sequences.

Abstract

We study the recursions $A(n) = A(n-a-A^k(n-b)) + A(A^k(n-b))$ where $a \geq 0$, $b \geq 1$ are integers and the superscript $k$ denotes a $k$-fold composition, and also the recursion $C(n) = C(n-s-C(n-1)) + C(n-s-2-C(n-3))$ where $s \geq 0$ is an integer. We prove that under suitable initial conditions the sequences $A(n)$ and $C(n)$ will be defined for all positive integers, and be monotonic with their forward difference sequences consisting only of 0 and 1. We also show that the sequence generated by the recursion for $A(n)$ with parameters $(k,a,b) = (k,0,1)$, and initial conditions $A(1) = A(2) = 1$, satisfies $A(E_n) = E_{n-1}$ where $E_n$ is defined by $E_n = E_{n-1} + E_{n-k}$ with $E_n = 1$ for $1 \leq n \leq k$.