Now I understand the question is about just H or just T single runs -?
The guessed formula isn't correct IMHO.
It can be solved either inductively or deductively. Who do you like more, Newton or Sherlock Holmes?
Single H runs in a sequence of n coin tosses consist of:
HT... at the beginning
.. THT … prior to the end
… TH at the end
Looking for a formula + derivation for the mean number of single H runs in a sequence of n tosses.
The proposed formula seems likely correct, given my numerics and small n cases. If so, all we need is the derivation. Whoever provides it shall be named honorary BD (see Monty Python link)