What is the number of possible outcomes of a flipped coin with the following property:

Let n be the number of times a coin is flipped, the outcome must be of the form that between two consecutive same side, there is an even number of the other side, or all are of one side....

For example

n=3,

The accepted outcome should be : TTT,HHH,TTH,HHT,HTT,THH. So I can't have an outcome like HTH or THT

n=4,

The accepted outcome should be : TTTT,HHHH,HTTH,THHT,HHTT,TTHH,TTTH, HHHT, HTTT, THHH.

n=5,

the accepted outcome should be: TTTTT, HHHHH, HTTHH, THHTT, TTHHT, HHTTH, TTTTH, TTTHH, HHHHT, HHHTT, HHTTT, TTHHH,HTTTT, THHHH .

So I know that there can only be 6, 10 and 14 possible outcomes for n = 3, 4 and 5 respectively for flipping a coin five time for this problem. How about for an arbitrary n.

I do hope you understand how I've explained this problem. Any reply would be greatly appreciated.