  An n-ary k-radius sequence is a finite sequence of elements taken from an
alphabet of size n such that any two distinct elements of the alphabet occur
within distance k of each other somewhere in the sequence. These sequences were
introduced by Jaromczyk and Lonc to model a caching strategy for computing
certain functions on large data sets such as medical images. Let f_k(n) be the
shortest length of any k-radius sequence. We improve on earlier estimates for
f_k(n) by using tilings and logarithms. The main result is that f_k(n) ~
n^2/(2k) as n tends to infinity whenever a certain tiling of Z^r exists. In
particular this result holds for infinitely many k, including all k < 195 and
all k such that k+1 or 2k+1 is prime. For certain k, in particular when 2k+1 is
prime, we get a sharper error term using the theory of logarithms.
