Motivated by the problem of finding explicit q-hypergeometric series which give rise to quantum modular forms, we define a natural generalization of Kontsevich's "strange" function. We prove that our generalized strange function can be used to produce infinite families of quantum modular forms. We do not use the theory of mock modular forms to do so. Moreover, we show how our generalized strange function relates to the generating function for ranks of strongly unimodal sequences both polynomially, and when specialized on certain open sets in C. As corollaries, we reinterpret a theorem due to Folsom-Ono-Rhoades on Ramanujan's radial limits of mock theta functions in terms of our generalized strange function, and establish a related Hecke-type identity.

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Given a recurrence sequence H, with H n = c 1 H n−1 + · · · + c t H n−t where c i ∈ N 0 for all i and c 1 , c t ≥ 1, the generalized Zeckendorf decomposition (gzd) of m ∈ N 0 is the unique representation of m using H composed of blocks lexicographically less than σ = (c 1 , . . . , c t ). We prove that the gzd of m uses the fewest number of summands among all representations of m using H, for all m, if and only if σ is weakly decreasing. We develop an algorithm for moving from any representation of m to the gzd, the analysis of which proves that σ weakly decreasing implies summand minimality. We prove that the gzds of numbers of the form v 0 H n +· · ·+v ℓ H n−ℓ converge in a suitable sense as n → ∞; furthermore we classify three distinct behaviors for this convergence. We use this result, together with the irreducibility of certain families of polynomials, to exhibit a representation with fewer summands than the gzd if σ is not weakly decreasing.

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We consider a problem first proposed by Mahler and Popken in 1953 and later developed by Coppersmith, Erdős, Guy, Isbell, Selfridge, and others. Let f (n) be the complexity of n ∈ Z + , where f (n) is defined as the least number of 1's needed to represent n in conjunction with an arbitrary number of +'s, * 's, and parentheses. Several algorithms have been developed to calculate the complexity of all integers up to n. Currently, the fastest known algorithm runs in time O(n 1.230175 ) and was given by J. Arias de Reyna and J. van de Lune in 2014. This algorithm makes use of a recursive definition given by Guy and iterates through products, f (d) + f n d , for d | n, and sums, f (a) + f (n − a), for a up to some function of n. The rate-limiting factor is iterating through the sums. We discuss potential improvements to this algorithm via a method that provides a strong uniform bound on the number of summands that must be calculated for almost all n. We also develop code to run J. Arias de Reyna and J. van de Lune's analysis in higher bases and thus reduce their runtime of O(n 1.230175 ) to O(n 1.222911236