Slow Recurrences

الأصل · EN

For positive integers α and β, we define an (α,β)-walk to be any sequence of positive integers satisfying wₖ₊₂=αwₖ₊₁+βwₖ. We say that an (α,β)-walk is n-slow if wₛ=n with s as large as possible. Slow (1,1)-walks have been investigated by several authors. In this paper we consider (α,β)-walks for arbitrary positive α,β. We derive a characterization theorem for these walks, and with this we prove several results concerning the total number of n-slow walks for a given n. In addition to this, we study the slowest n-slow walk for a given n amongst all possible α,β.