On generalized Tribonacci sequences and additive partitions (Q1567661)

A set \(U\) of positive integers is called avoidable if there exists a partition \(A\), \(B\) of all positive integers such that no element of \(U\) is a sum of two distinct elements of \(A\) or two distinct elements of \(B\). The author investigates the avoidability of the generalized Tribonacci sequences \(T= t_i\), which are defined by recurrence \(t_{n+3}=t_{n+2}+t_{n+1}+t_n\) with positive integral initial terms \((t_1, t_2, t_3)=(a, b, c)\). He finds necessary and sufficient conditions for \(T\) avoidable in two cases: (1) \(a<b<c< a+b\); (2) \(a<b<c, a+b<c\) and \(c=d\pmod {a+b}\) with \(b-a-1<d<a+b\). The results extend the family of known avoidable sets given earlier by Hoggatt jun., Shan and Zhu (see the references of the paper). The tools in the proofs mainly are elementary number theory and graph theory. The authors remark that recently \textit{M. Develin} [Electron. J. Comb. 7, No. 1, R53 (2000; Zbl 0964.05008)] has obtained a complete result to answer whether a generalized Tribonacci sequence with any positive integral initial terms is avoidable.