A Complete Categorization of When Generalized Tribonacci Sequences Can Be Avoided by Additive Partitions (original) (raw)

Abstract

A set or sequence UUU in the natural numbers is defined to be avoidable if bfN{\bf N}bfN can be partitioned into two sets AAA and BBB such that no element of UUU is the sum of two distinct elements of AAA or of two distinct elements of BBB. In 1980, Hoggatt [5] studied the Tribonacci sequence T=tnT=\{t_n\}T=tn where t1=1t_1=1t1=1, t2=1t_2=1t2=1, t3=2t_3=2t3=2, and tn=tn−1+tn−2+tn−3t_n=t_{n-1}+t_{n-2}+t_{n-3}tn=tn−1+tn−2+tn−3 for nge4n\ge 4nge4, 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.