Van Lier sequences

We study two types of sequences of positive integers which arise from problems in the measurement of comparative judgements of probability. The first type consists of the Van Lier sequences, which are nondecreasing sequences x₁, x₂,...,x_n of positive integers that start with two 1's and have the property that, whenever j<k≤n,x_k-x_j can be expressed as a sum of terms from the sequence other than x_j. The second type consists of the regular sequences, which are nondecreasing sequences of positive integers that start with two 1's and have the property that each subsequent term is a partial sum of preceding terms. We show that every regular sequence without "gaps" is Van Lier and that every regular sequence which satisfies the Fibonacci-like inequality x_k≤x_k-2+x_k-1 is Van Lier. We also study one-term extensions of Van Lier sequences and obtain some asymptotic results on the number of Van Lier sequences.