We agree that we can eliminate rows 1 and 2 from the further analysis. In rows 3–6 (2/3 of the rows) heads follows heads 25% of the time. In rows 7–8 (1/3 of the rows), heads follows heads 75% of the time. So over the six rows, heads follows heads 25% x 2/3 + 75% x 1/3 = 50%/3 + 25% = approximately 42%.
Here is a thought experiment.
Imagine we play a game in which, in each round, a coin is flipped three times in a row. Each round you and I pay 50 cents to play, so there is a pot of $1. If the proportion of heads following heads is 100%, you win the pot, if the proportion of tails following heads is 100%, I win the pot. If the proportion is 50% (as in row 7), we receive our stake back. (And if there is no opportunity for anything to follow heads, our stake rolls over to the next round).
If you are correct, in the long run, we will both break even (which is still better odds than playing the lottery :-)). Will you play this game with me?