Generating Function for 0,1 Sequences with Period m

[Emilijo]

What is generating function of these sequences:
0,1,0,1...or
0,1,1,0,1,1,0...or
0,1,1,1,0,1,1,1,0...

where m is period, in first example m=2; in the second m=3, and so on.
Generating function must be general, so I can just put m.

 

[epsi00]

look up your sequences here:

https://oeis.org/

if they are known, they will be listed.

 

[Emilijo]

I am not satisfied. Can anyone answer on my question?

 

[micromass]

So you need to find a function such that

$$f(x)=1+x^2+x^4+x^6+x^8+...$$

Do you know the function g such that

$$g(x)=1+x+x^2+x^3+x^4+x^5+...$$

Then $f(x)=g(x^2)$...

 

[fbs7]

So you need to find a function such that

$$f(x)=1+x^2+x^4+x^6+x^8+...$$

Do you know the function g such that

$$g(x)=1+x+x^2+x^3+x^4+x^5+...$$

Then $f(x)=g(x^2)$...

The other ones should be the same way, right? Like

$$f(x) = x + x^2 + x^4 + x^5 + x^7 + x^8 + ... = (1 + x + x^2 + x^3 + x^4 + ... ) - (1 + x^3 + x^6 + x^9 + ...) = g(x) - g(x^3)$$

 

[micromass]

Indeed! That how I'd do the other ones!

 

[fbs7]

Indeed! That how I'd do the other ones!

You're a great mentor! I had never heard of generating functions until I came to this post, but your post got me curious to go find the mechanics around that.

Learning a new thing every day. Thank you! :D

 

[micromass]

You're a great mentor! I had never heard of generating functions until I came to this post, but your post got me curious to go find the mechanics around that.

Learning a new thing every day. Thank you! :D

If you're really interested then I can recommend the book "Concrete Mathematics" by Knuth. The book gives several mathematical tools to solve problems that sound easy but whose solutions are far from it! Generating functions is one of those tools.

 

[Xitami]

wee ned a proof for$$\overbrace{\,0,\,1,...\,1}^m\quad\frac{\sum_{i=1}^{m-1}x^i}{1\,-\,x^m}$$