Elemantary Sequences, Sub-Fibonacci Sequences

Abstract Recent research in uniqueness of representability for finite measurement structures has identified a number of novel finite integer sequences. One of the simplest, called an elementary sequence, is a nondecreasing integer sequence x 1 , x 2 ,…, x n with x 1 = x 2 =1 and, for all k > 2, if x k > 1 then x k = i + j for distinct i , j k . We investigate combinatorial and number-theoretic questions for elementary sequences and identify interesting open problems. The paper also discusses sub-Fibonacci sequences x 1 , x 2 ,…, x n , which are characterized as nondecreasing integer sequences with x 1 = x 2 =1 and x k ≤ x k −1 + x k −2 for each k > 2.