Formula for this combination problem

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SUMMARY

The discussion focuses on determining the number of valid outcomes when flipping a coin 'n' times, where the outcomes must adhere to specific conditions. For n=3, the valid outcomes are TTT, HHH, TTH, HHT, HTT, and THH, totaling 6 outcomes. For n=4, the valid outcomes increase to 10, including combinations like HTTH and THHT. For n=5, there are 14 valid outcomes, such as HTTHH and TTHHT. The key rule is that between two consecutive same sides, there must be an even number of the opposite side, or all outcomes can consist of one side only.

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rhonkie
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Hi,

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.
 
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I don't really understand it... TTH does not have an even number of one side between two of the other side like THHT does, for example; nor does is consist of three of the same side like TTT does. So why again is this allowed?
 
Well, one can't really see this allowable pattern for n less than or equal to 3. Perhaps what I should have said is that the following sort of outcome is not allowed:

between two consecutive same sides, odd number of the other sides is not counted as accepted outcome. Every other outcomes are allowed.

Sorry for the ambiguity in my post. And thanks in advance
 

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