  A palindromic composition of $n$ is a composition of $n$ which can be read
the same way forwards and backwards. In this paper we define an
anti-palindromic composition of $n$ to be a composition of $n$ which has no
mirror symmetry amongst its parts. We then give a surprising connection between
the number of anti-palindromic compositions of $n$ and the so-called tribonacci
sequence, a generalization of the Fibonacci sequence. We conclude by defining a
new q-analogue of the Fibonacci sequence, which is related to certain
equivalence classes of anti-palindromic compositions
