Sums of Ceiling Functions Solve Nested Recursions

TL;DR

This paper demonstrates that certain nested recursions can be explicitly solved using sums of ceiling functions, providing closed-form solutions and combinatorial interpretations for specific parameter conditions.

Contribution

It introduces a novel approach to solving nested recursions with ceiling functions and establishes necessary and sufficient conditions for their solutions.

Findings

Closed-form solutions for specific nested recursions using ceiling sums

Characterization of parameters for which ceiling sums solve the recursions

Combinatorial interpretations via labeled trees

Abstract

It is known that, for given integers s \geq 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). Further, given appropriate initial conditions, we derive necessary and sufficient conditions on the parameters s1, a1, s2 and a2 so that C(n) solves the nested recursion R(n) = R(n - s1 - R(n - a1)) + R(n- s2 - R(n - a2)).