Nested recursions with ceiling function solutions

TL;DR

This paper investigates nested recursions with ceiling function solutions, develops an algorithm to identify when such functions are solutions, and finds that ceiling{n/q} solutions are common, especially when r=1.

Contribution

The authors introduce an algorithm to determine when ceiling functions solve complex nested recursions and establish that ceiling{n/q} solutions occur infinitely often, with evidence supporting a conjecture about their form.

Findings

Every ceiling{n/q} is a solution to infinitely many such recursions.

Empirical evidence suggests solutions only occur when r=1.

Connected recursion to a generalization of Conway's recursion.

Abstract

Consider a nested, non-homogeneous recursion R(n) defined by R(n) = \sum_{i=1}^k R(n-s_i-\sum_{j=1}^{p_i} R(n-a_ij)) + nu, with c initial conditions R(1) = xi_1 > 0,R(2)=xi_2 > 0, ..., R(c)=xi_c > 0, where the parameters are integers satisfying k > 0, p_i > 0 and a_ij > 0. We develop an algorithm to answer the following question: for an arbitrary rational number r/q, is there any set of values for k, p_i, s_i, a_ij and nu such that the ceiling function ceiling{rn/q} is the unique solution generated by R(n) with appropriate initial conditions? We apply this algorithm to explore those ceiling functions that appear as solutions to R(n). The pattern that emerges from this empirical investigation leads us to the following general result: every ceiling function of the form ceiling{n/q}$ is the solution of infinitely many such recursions. Further, the empirical evidence suggests that the converse conjecture is true: if ceiling{rn/q} is the solution generated by any recursion R(n) of the form above, then r=1. We also use our ceiling function methodology to derive the first known connection between the recursion R(n) and a natural generalization of Conway's recursion.