For $k>1$ and nonnegative integer parameters $a_p, b_p$, $p = 1..k$, we analyze the solutions to the meta-Fibonacci recursion $C(n)=\sum_{p=1}^k C(n-a_p-C(n-b_p))$, where the parameters $a_p, b_p$, $p = 1..k$ satisfy a specific constraint. For $k=2$ we present compelling empirical evidence that solutions exist only for two particular families of parameters; special cases of the recursions so defined include the Conolly recursion and all of its generalizations that have been studied to date. We show that the solutions for all the recursions defined by the parameters in these families have a natural combinatorial interpretation: they count the number of labels on the leaves of certain infinite labeled trees, where the number of labels on each node in the tree is determined by the parameters. This combinatorial interpretation enables us to determine various new results concerning these sequences, including a closed form, and to derive asymptotic estimates. Our results broadly generalize and unify recent findings of this type relating to certain of these meta-Fibonacci sequences. At the same time they indicate the potential for developing an analogous counting interpretation for many other meta-Fibonacci recursions specified by the same recursion for $C(n)$ with other sets of parameters.

A nondecreasing sequence of positive integers is (α, β)-Conolly, or Conollylike for short, if for every positive integer m the number of times that m occurs in the sequence is α + βr m , where r m is 1 plus the 2-adic valuation of m. A recurrence relation is (α, β)-Conolly if it has an (α, β)-Conolly solution sequence. We discover that Conolly-like sequences often appear as solutions to nested (or meta-Fibonacci) recurrence relations of the formA(n − a ij )) with appropriate initial conditions. For any fixed integers k and p 1 , p 2 , . . . , p k we prove that there are only finitely many pairs (α, β) for which A(n) can be (α, β)-Conolly. For the case where α = 0 and β = 1, we provide a bijective proof using labelled infinite trees to show that, in addition to the original Conolly recurrence, the recurrence H(n) = H(n − H(n − 2)) + H(n − 3 − H(n − 5)) also has the Conolly sequence as a solution. When k = 2 and p 1 = p 2 , we construct an example of an (α, β)-Conolly recursion for every possible (α, β) pair, thereby providing the first examples of nested recursions with p i > 1 whose solutions are completely understood. Finally, in the case where k = 2 and p 1 = p 2 , we provide an if and only if condition for a given nested recurrence A(n) to be (α, 0)-Conolly by proving a very general ceiling function identity.

Abstract. The solutions to certain nested recursions, such as Conolly's C(n) = C(n − C(n − 1)) + C(n − 1 − C(n − 2)), with initial conditions C(1) = 1, C(2) = 2, have a well-established combinatorial interpretation in terms of counting leaves in an infinite binary tree. This tree-based interpretation, which has a natural generalization to a k-term nested recursion of this type, only applies to homogeneous recursions, and only solves each recursion for one set of initial conditions determined by the tree. In this paper, we extend the tree-based interpretation to solve a non-homogeneous version of the k-term recursion that includes a constant term. To do so we introduce a tree-grafting methodology that inserts copies of a finite tree into the infinite k-ary tree associated with the solution of the corresponding homogeneous k-term recursion. This technique can also be used to solve the given non-homogeneous recursion with various sets of initial conditions.

No abstract

We define the generalized Golomb triangular recursion by g j,s,λ (n) = g j,s,λ (n − s−g j,s,λ (n−j))+λj. 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 λ = 1 the initial conditions for these alternative tree interpretations coincide and we derive explicit closed forms for the solution sequence and its frequency function.