<s>[INST] Writing $1+3x^2+8x^4+21x^6+\cdots$ as a power series representation How would I write the power series $$1+3x^2+8x^4+21x^6+\cdots$$ as a power series  representation (something neat similar to $\frac{1}{1-x}$)? 
This reminds me of the power series $1+x^2+x^4+x^6+\cdots$ where the power series representation for that is $\frac{1}{1-x^2}$, but how would I add the Fibonacci numbers as coefficients into that?
Hints only!!
 [/INST] Let $f(x) = 1+2x+3x^2+5x^3+8x^4+13x^5+21x^6+\cdots$. 
You probably already know a closed form for $f(x)$. 
Then, $f(-x) = 1-2x+3x^2-5x^3+8x^4-13x^5+21x^6-\cdots$.
Do you see how to get the series you want from $f(x)$ and $f(-x)$?

To get a closed form for $f(x)$ try combining the following equations in a way that leaves a finite number of terms on the right side: 
$f(x) \ \ \ \ = 1+2x+3x^2+5x^3+8x^4+13x^5+21x^6+\cdots$
$xf(x) \ \ = \ \ \ \ \ \ \ 1x+2x^2+3x^3+5x^4+ \ \ 8x^5+13x^6+\cdots$
$x^2f(x) = \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1x^2+2x^3+3x^4+ \ \ 5x^5+ \ \ 8x^6+\cdots$
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