AbstractThis paper investigates solutions of the general recurrence M(0) = g(0), M(n + 1) = g(n + 1) + min0⩽k⩽n(αM(k) + βM(n − k)) for various choices of α, β, and g(n). In a large number of cases it is possible to prove that M(n) is a convex function whose values can be computed much more efficiently than would be suggested by the defining recurrence. The asymptotic behavior of M(n) can be deduced using combinatorial methods in conjunction with analytic techniques. In some cases there are strong connections between M(n) and the function H(x) defined by H(x) = 1 for x < 1, H(x) = H((x − 1)α) + H((x − 1)β) for x ⩾ 1. Special cases of these recurrences lead to a surprising number of interesting problems involving both discrete and continuous mathematics.