A complete categorization of when generalized Tribonacci sequences can be avoided by additive partitions (Q1585654)

Let \(C\) be a set of positive integers. \(C\) is said to be avoidable if there is a partition of \(\mathbb{N}\) into two parts \(A\) and \(B\) such that \(A+B\cap C=\emptyset\), where \(A+B=\{a+b \mid a\in A\), \(b\in B\}.\) For example the Fibonacci sequence is avoidable as Alladi, Erdős and Hoggatt proved. Hoggatt, and later Dumitru, investigated the sequence with recursion \(s_n=s_{n-1}+s_{n-2}+s_{n-3}\) and special initial terms. In the present paper the author investigates the case when the initial terms are arbitrary. He establishes a finite set \(C\) showing that \(\{s_n\}\) is avoidable if and only if \(C\) is avoidable.