Sums of Ceiling Functions Solve Nested Recursions

Abstract

It is known that, for given integers s ≥ 0 and j > 0, the nested recursion R(n) = R(n−s−R(n− j))+R(n−2j−s−R(n−3j)) has a closed form solution for which a combinatorial interpretation exists in terms of an infinite, labeled tree. For s = 0, we show that this solution sequence has a closed form as the sum of ceiling functions C(n) = ∑j−1 i=0 ⌈ 

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