#A17 INTEGERS 12A (2012): John Selfridge Memorial Issue SOME MONOAPPARITIC FOURTH ORDER LINEAR DIVISIBILITY SEQUENCES

A sequence of rational integers {An} is said to be a divisibility sequence if Am | An whenever m | n. If the divisibility sequence {An} also satisfies a linear recurrence relation of order k, it is said to be a linear divisibility sequence. The best known example of a linear divisibility sequence of order 2 is the Lucas sequence {un}, one particular instance of which is the famous Fibonacci sequence. In their extension of the Lucas functions to order 4 linear recursions, Williams and Guy showed that the order 4 analog {Un} of {un} can have no more than two ranks of apparition for a given prime p and frequently has two such ranks, unlike the situation for {un}, which can only have one rank of apparition. In this paper we investigate the problem of finding those sequences {Un} which have only one rank of apparition for any prime p. – In memory of John Selfridge, a close friend and collaborator for nearly half a century.