Sum-distinct sequences and Fibonacci numbers

Abstract

Positive integers a, ~2” holds for every sumdistinct sequence. When one wants to create sum-distinct sequences A,, n = 1, 2, . . . , with small elements, the first thought consists in trying to extend A,-, with the smallest possible integer which is not the subsum of elements in AnO1. Not too surprisingly, this “first-fit” approach works when one starts with A, = {l},anditresultsinthesequenceof 2powersA, = (2’: 0 <i In 1) for each n. By using more sophisticated greedy procedures, one gets “better” sumdistinct sequences, that is sequences with smaller maximal element. It is worth noting that the sequence given by Conway and Guy in [2] (c.f. p. 64 in [6]), and which they conjecture answers the problem of ErdSs and Moser, is a greedy-like construction as well. The introduction of the irregularity strength of graphs by Chartrand et al. in [l] led to extend the concept of a sum-distinct sequence to more than one sequence and their generation by “greedy” procedures analogous to the first-fit approach.