  We show that the number of length-n words over a k-letter alphabet having no
even palindromic prefix is the same as the number of length-n unbordered words,
by constructing an explicit bijection between the two sets. A slightly
different but analogous result holds for those words having no odd palindromic
prefix. Using known results on borders, we get an asymptotic enumeration for
the number of words having no even (resp., odd) palindromic prefix . We obtain
an analogous result for words having no nontrivial palindromic prefix. Finally,
we obtain similar results for words having no square prefix, thus proving a
2013 conjecture of Chaffin, Linderman, Sloane, and Wilks.
