In an unpublished problem collection, Motzkin asks, how dense can a set S of positive integers be, if no two elements of S are allowed to di?er by an element of the given set P of positive integers? The maximal density of such sets, denoted by μ(P), is known for |P| ? 2, and several other partial results are also known for the general case. We find some bounds and a few exact values of μ(P), where the elements Pi of the set P are defined by Pi := Pi1+Pi2, i 2 with P0 = a, P1 = b. Notice that the elements of the sequence {Pi} satisfy the same recurrence relation as that satisfied by the well-known Fibonacci numbers Fi with arbitrary initial values. Since Pi = aFi1 + bFi for all i 0, these numbers are also known as weighted Fibonacci numbers. This work generalizes an earlier work of Pandey on Fibonacci numbers.
展开▼
