Nested Recurrence Relations with Conolly-like Solutions

A non-decreasing 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 + βrm, where rm 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 form A(n) = ∑k i=1 A(n− si− ∑pi j=1 A(n− aij)) with appropriate initial conditions. For any fixed integers k and p1, p2, . . . , pk 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 p1 = p2, 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 pi > 1 whose solutions are completely understood. Finally, in the case where k = 2 and p1 = p2, 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.