Find closed formula for the sequence.

[jjangub]

I have a problem with this question.
The question is "Find closed formula for the sequence 0,1,3,0,1,3,0,1,3..."
It can be written as (0,1,0,0,1,0,0,1,0...)+3*(0,0,1,0,0,1,0,0,1...)
I know the sequence of 0,0,1,0,0,1..., but I do not know how to get the
sequence of 0,1,0,0,1,0...
And is there any easier way to get the sequence of 0,0,1,0,0,1...or 0,1,0,0,1,0...?
Thank you...

 

[CRGreathouse]

I have a problem with this question.
The question is "Find closed formula for the sequence 0,1,3,0,1,3,0,1,3..."
It can be written as (0,1,0,0,1,0,0,1,0...)+3*(0,0,1,0,0,1,0,0,1...)
I know the sequence of 0,0,1,0,0,1..., but I do not know how to get the
sequence of 0,1,0,0,1,0...
And is there any easier way to get the sequence of 0,0,1,0,0,1...or 0,1,0,0,1,0...?
Thank you...

Depends on what you're allowed to use, and what techniques you know. In Pari it would be

[3,0,1][n%3+1]

. The "mod" function would make quick work of it. Complex powers could work: with w a cube root of -1, $$w^k+\bar{w^k}$$ is 1, -1, or 2 depending on k mod 3. Etc.

Edit: The conjugate symbol isn't showing up very well...