The correct answer is 68. The expected value of any pattern W of length m, comprised of any symbols, W=w1w2..wm, is given by the following formula:
E[T]=[1/p(W)][1+Sum p(k) over all k such that k is a period of W],
where p(k)=p(w1)p(w2)...p(wk), and P(W)=p(m). In coin tossing experiments, only two symbols are involved, H and T. If W = HHTTHH, then W has only one period, k=4. Assuming a fair coin (which need not be assumed in using the formula),
p(k)= p(4)=1/(2^4) and 1/p(W) = 2^6. Substituting these values into the above formula gives
E[T] = 2^6[1 + 1/(2^4)] = 2^6[17/(2^4)] = (2^2)(17) = 68.