A combinatorial approach for solving certain nested recursions with non-slow solutions

TL;DR

This paper introduces a generalized recursive sequence defined by a specific recurrence relation, explores its solutions with combinatorial interpretations via labeled trees, and derives explicit formulas for certain cases.

Contribution

It presents a new combinatorial approach to solving nested recursions with non-slow solutions, including explicit formulas and multiple tree interpretations.

Findings

Solutions are non-slow monotone sequences with combinatorial interpretations.

Multiple tree interpretations lead to different solutions for the recursion.

Explicit closed-form solutions are derived for the case when lambda=1.

Abstract

We define the generalized Golomb triangular recursion by g_{j,s,lambda}(n) = g_{j,s,lambda}(n - s - g_{j,s,lambda}(n-j)) + \lambda j. For particular choices of the initial conditions, we show that the solution of the recursion is a non-slow monotone sequence for which we can provide a combinatorial interpretation in terms of a weighted count of the leaves of a certain labeled infinite tree. We discover that more than one such tree interpretation is possible, leading to different choices of the initial conditions and alternative solutions that are closely related. In the case lambda=1 the initial conditions for these alternative tree interpretations coincide and we derive explicit closed forms for the solution sequence and its frequency function.