Generating Function & Sequence

Question

Find the generating functions of the sequences 2, 1, 2, 1, 2, 1, . . .

I get $\frac{1}{1+x} + \frac{1}{1-x} = \frac{2}{1-x^2}$

But the solution ends up with $\frac{2}{1-x^2} + \frac{x}{1-x^2} = \frac{2+x}{1-x^2}$. The solutions starts with $\sum_{n\ge 0} (2)x^{2n}+\sum_{n\ge 0} (1) x^{2n+1}$

I couldn't come up with anything like that. I feel like I'm confused with something.

Answer 1

As $\frac{1}{1-x}$ is the generating series for $1,1,1,\dots$ and $\frac{1}{1-x}$ is the generating series for $1,-1,1,-1,\dots$, the series for which you are calculating the generating function for is $2,0,2,0,\dots$.

But we can get the result you want by noticing that $2,1,2,1,\dots$ is actually $\frac{3}{2},\frac{3}{2},\frac{3}{2},\dots$ plus $\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\dots$.

We know the first series has generating function $\frac{3/2}{1-x}$ and the second has generating function $\frac{1/2}{1+x}$. Thus, the function you are looking for is$$\frac{\frac{3}{2}}{1-x}+\frac{\frac{1}{2}}{1+x}=\frac{\frac{3}{2}(1+x)+\frac{1}{2}(1-x)}{(1-x)(1+x)}=\frac{2+x}{1-x^2}$$ which is the result you were given.

Answer 2

As an alternative to the other answers, you can simply use a variant of one function, that being

$$\frac{1}{1-x^2}=\sum_{n=0}^{\infty} x^{2n}$$

If you consider that the $2$'s are even indexed (assuming in the sequence $\{a_n\}$ the first term is $a_0$) and the $1$'s are odd indexed, you have that

$$\frac{2}{1-x^2}$$

will generate the $2$'s and

$$\frac{x}{1-x^2}$$

will generate the $1$'s. This is because

$$\frac{1}{1-x^2}=1+x^2+x^4+...$$

and so

$$\frac{x}{1-x^2}=x(1+x^2+x^4+...)=x+x^3+x^5+...$$

Thus, you sequence will be generated by

$$f(x)=\frac{2+x}{1-x^2}$$

Answer 3

The sequence $2,1,2,1,2,1,...$ alternates between $2$ and $1$, being $2$ for even-numbered terms and $1$ for odd-numbered terms. The generating function is thus $$\sum\limits_{n \ge0}(2)x^{2n}+\sum\limits_{n\ge0}(1)x^{2n+1}=\sum\limits_{k\ge0}(1)x^k+\sum\limits_{k\ge0}(1)x^{2k}=\dfrac{1}{1-x}+\dfrac{1}{1-x^2}=\dfrac{2+x}{1-x^2}$$

Answer 4

And if you want to generate $(a_i)_{i=1}^m$ repeatedly, the i-th term is generated by $a_ix^i$, and for this to repeat every $m$ this needs $\dfrac{a_ix^i}{1-x^m} $ so the final result is $\sum_{i=1}^m\dfrac{a_ix^i}{1-x^m} $.

In your case, with $(1, 2)$ repeated, this is $\dfrac{x}{1-x^2}+\dfrac{2x^2}{1-x^2} =\dfrac{x+2x^2}{1-x^2} $.

This starts the first term at $x^1$. To start it at the constant term, just reduce the exponents of the $x^i$ by one giving $\sum_{i=1}^m\dfrac{a_ix^{i-1}}{1-x^m} $, so this gives, in this case $\dfrac{1}{1-x^2}+\dfrac{2x}{1-x^2} =\dfrac{1+2x}{1-x^2} $.