This paper concerns the problem of determining or estimating the maximal upper density of the sets of nonnegative integers *S* whose elements do not differ by an element of a given set *M* of positive integers. We find some exact values and some bounds for the maximal density when the elements of *M* are generalized Fibonacci numbers of odd order. The generalized Fibonacci sequence of order *r* is a generalization of the well known Fibonacci sequence, where instead of starting with two predetermined terms, we start with *r* predetermined terms and each term afterwards is the sum of *r* preceding terms. We also derive some new properties of the generalized Fibonacci sequence of order *r*. Furthermore, we discuss some related coloring parameters of distance graphs generated by the set *M*.

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