Find series which of generating function is $A(z) = z^{3} A(z) +1 $ and equation for n'th element of series.
I wanted to check wether my solution is right . Series for z^3 is (0,0,0,1,0,...) and series for z^3 + 1 is (1,0,0,1,0,0,...) so equation for n'th element of series is $a_{n} = [n=0] + [n=3]$ is that correct?
Put $$A(z) = \sum_{n=0}^\infty a_nz^n.$$ Then you have $$A(z) = 1 + z^3A(z),$$ so $$\sum_{n=0}^\infty a_nz^n = 1 + \sum_{n=0}^\infty a_nz^{n+3}.$$ Equating the terms in the two power series gives $a_0 = 1$, $a_1 = a_2 = 0$. After that, you see $a_n = a_{n-3}$ for $n\ge 3$. This yields $a_{3n} = 1$ for all $n$ and the rest of the coefficients are zero.
Sub everthing in and watch it work.