What is the generating function for $1,1,1,1,1,1$?
I know this to be $1 + x +x^2+ x^3+x^4+x^5$
But then I saw this:
$$\frac{x^6-1}{x-1} = 1 + x +x^2+ x^3+x^4+x^5$$
How was this equality obtained?
Was it just a random (manual)? or is there any method involved to obtain that fractional part?
As noted in the comments to your question, the equation
$$\frac{x^6 - 1}{x-1} = 1 + x + x^2 + x^3 + x^4 + x^5$$
comes about as the sum of a finite geometric series. Suppose we have a finite geometric series of ratio $x$. Then it can be shown
$$1 + x + x^2 + ... + x^n = \frac{x^{n+1} - 1}{x-1}$$
Take $n=5$ and the equality results.