Integer sequences with given sum and restricted differences

Abstract

Let d₊, d⁺ be positive integers with d₊≤ d⁺. We investigate the possibility of writing a positive integer n as the sum of an increasing sequence of positive integers such that the first term is at most d⁺ and the difference between consecutive terms lies between d⁺ and d₊. We establish a recurrence relation to enumerate the sequences with given sum and last term and derive methods for constructing all such sequences with a given sum.