Given a recurrence sequence H, with H n = c 1 H n−1 + · · · + c t H n−t where c i ∈ N 0 for all i and c 1 , c t ≥ 1, the generalized Zeckendorf decomposition (gzd) of m ∈ N 0 is the unique representation of m using H composed of blocks lexicographically less than σ = (c 1 , . . . , c t ). We prove that the gzd of m uses the fewest number of summands among all representations of m using H, for all m, if and only if σ is weakly decreasing. We develop an algorithm for moving from any representation of m to the gzd, the analysis of which proves that σ weakly decreasing implies summand minimality. We prove that the gzds of numbers of the form v 0 H n +· · ·+v ℓ H n−ℓ converge in a suitable sense as n → ∞; furthermore we classify three distinct behaviors for this convergence. We use this result, together with the irreducibility of certain families of polynomials, to exhibit a representation with fewer summands than the gzd if σ is not weakly decreasing.
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