Two Patterns, Two Clocks

Keep flipping a fair coin. Let $a$ be the expected number of flips until you first see a head immediately followed by another head (HH), and let $b$ be the expected number until you first see a head immediately followed by a tail (HT). In any two fixed flips the patterns HH and HT are equally likely — yet $a \neq b$. Compute both expectations and explain the asymmetry.