Sums of ceiling functions solve nested recursions

Abstract

It is known that, for given integers j S 0 and j > 0, the nested recursion R{n) = R{n -s -R(n -j)) -\-R(n -2j -s -R{n -2/)) has a closed-form solution for which a combinatorial interpretation exists in terms of an infinite, labelled tree. For 5 = 0, we show that this solution sequence has a closed form as the sum of ceiling functions Furthermore, given appropriate initial conditions, we derive necessary and sufficient conditions on the parameters S\,a\,S2 and 02 so that C(n) solves the nested recursion R(n) = R(n -ii -R{n -a\)) + R{n -si -R{n -ai)).