On Tribonacci Sequences

TL;DR

This paper investigates the properties of tribonacci sequences, establishing a finite upper bound on the number of such sequences with given constraints and analyzing their transition behavior around specific exponential bounds.

Contribution

The authors prove a finite upper bound on the number of tribonacci sequences with positive initial terms for any fixed term and analyze the transition points based on exponential functions involving the tribonacci constant.

Findings

Finite upper bound of 561001 on the number of sequences for any n≥3.

Transition points in n occur around multiples of ^{3k/2} involving the tribonacci constant.

Values of f_k(n) are bounded independently of k below a certain exponential threshold.

Abstract

Let a tribonacci sequence be a sequence of integers satisfying $a_k=a_{k-1}+a_{k-2}+a_{k-3}$ for all $k\ge 4$. For any positive integers $k$ and $n$, denote by $f_k(n)$ the number of tribonacci sequences with $a_1, a_2, a_3>0$ and with $a_k=n$. For all $n$, there is a maximum $k$ such that $f_k(n)$ is non-zero. Answering a question of Spiro \cite{Spiro}, we show that there is a finite upper bound (we specifically prove 561001) on $f_k(n)$ for any positive integer $n\ge 3$ and this maximum $k$. We do this by showing that $f_k(n)$ has transitions in $n$ around constant multiples of $\phi^{3k/2}$ (where $\phi$ is the real root of $\phi^3=\phi^2+\phi+1$): there exists a constant $C$ such that $f_k(n)>0$ whenever $n>C\phi^{3k/2}$ and for any constant $T$, the values of $f_k(n)$ with $n<T\phi^{3k/2}$ have an upper bound independent of $k$.