To find pattern(if it exists) for finite sequences

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I am working on something in whose outputs I would be getting pairs of finite sequences (with entries from the set {1,-1} and starting with 1) of length n(n>1). Now, I need to find a general pattern for such pairs corresponding to n.

For n=2, I obtained the pair: {(1,-1) ; (1,1)}

For n=4, I obtained the pair: {(1,-1,-1,-1) ; (1,1,-1,1)}

For n=8, I obtained four sets of pairs:{ (1,-1,-1,1,1,1,-1,-1) ; (1,-1,1,1,1,1,1,-1)} , { (1,-1,-1,1,1,1,-1,-1) ; (1,1,1,-1,1,-1,1,1)}, { (1,-1,1,1,1,1,1,-1) ; (1,1,-1,-1,1,-1,-1,1)}, { (1,1,-1,-1,1,-1,-1,1) ; (1,1,1,-1,1,-1,1,1)}

For n=3,5,6,7,9,10, no such pairs were found.

For n=10, I had to run the code on a supercomputer. For n=11 ,the code ran for one week and it was aborted,as there is time limit of one week to run the codes on the supercomputer to which I have access .

It seems that outputs are obtained when n is a power of 2.

From the obtained outputs, is there a way to find a pattern so that for n=16 etc. the required pairs could be found.Then , I will see , if it works fine with the thing I am working on.