Coin toss

To satisfy the above condition, the last four flips must be an exact reflection of the first four (The first four can be anything).

There are 16 possible outcomes for four coin tosses and only 1 can be a reflection.

Consider the following sequence, where the top row denotes a possibility of heads and the bottom denotes tails:

1 2 3 4 5 6 7 8 H H H H H H H H T T T T T T T T T

To get a palindrome sequence, we can ignore the last 4, since they must reflect the first 4. So we only consider these four like so:

1 2 3 4 H H H H T T T T

There are two choices each for each of the four slots, so that is $2^4$ . The above arrangement has $2^8$ possibilities. $2^4$ / $2^8$ = 1/ $2^4$ = 1/16 which is a/b. So a + b = 1 + 16 = $\boxed{17}$ .