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

A set or sequence U in the natural numbers is defined to be avoidable if N can be partitioned into two sets A and B such that no element of U is the sum of two distinct elements of A or of two distinct elements of B. In 1980, Hoggatt [5] studied the Tribonacci sequence T = {tn} where t1 =1,t2 =1,t3 =2,and tn = tn−1 + tn−2 + tn−3 for n ≥ 4, and showed that it was avoidable. Dumitriu [3] continued this research, investigating Tribonacci sequences with arbitrary initial terms, and achieving partial results. In this paper we give a complete answer to the question of when a generalized Tribonacci sequence is avoidable