Generating function , closed form

Question

Find a closed form for the generating function for the given sequence

1) 1,1,0,1,1,1,1,1.....

2) 1,2,1,1,1,1,1.....

Do manually​ ..

Answer 1

$1$

$\begin{align*} &\Rightarrow 1,1,0,1,1,1,1,1,1,1 \; .... \infty \\ &\Leftrightarrow 1.x^0 + 1.x^1 + {\color{red}{0.x^2}} + 1.x^3 + 1.x^4 + ... \infty \\ &\Leftrightarrow \left ( 1.x^0 + 1.x^1 + {\color{green}{1.x^2}} + 1.x^3 + 1.x^4 + ...\infty \right ) - {\color{red}{1.x^2}} \\ &\Leftrightarrow \frac{1}{1-x} - 1.x^2 \\ &\Leftrightarrow \frac{1-x^2+x^3}{1-x} \\ \end{align*}$

$2$

$\begin{align*} &\Rightarrow 1,2,1,1,1,1,1,1,1,1 \; .... \infty \\ &\Leftrightarrow 1.x^0 + 2.x^1 + 1.x^2 + 1.x^3 + 1.x^4 + ... \infty \\ &\Leftrightarrow \left ( 1.x^0 + 1.x^1 + 1.x^2 + 1.x^3 + 1.x^4 + ...\infty \right ) +1.x^1 \\ &\Leftrightarrow \frac{1}{1-x} + 1.x^1 \\ &\Leftrightarrow \frac{1+x-x^2}{1-x} \\ \end{align*}$

Comments

Thanks for verify  ,I am getting same ....yeah :)