Find the generating function of $S_n$(Please check my idea)

$Q)$ Let the $S_n$ be the number of the positive roots $(x,y,z)$ of the $x+2y+3z=n$, $n geq 6$ (I.e. $x>0, y>0$ and $z>0$)

Find the generating function $g(x)$ of $S_n + S_{n+1} + S_{n+2}$

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Say $f(x) = S_6 x^6 +S_7x^7+ S_8x^8…$ (generating function of the $S_n$)

When I computed the $f(x)$, $f(x)$ = $x^6 over (1-x)(1-x^2)(1-x^3)$

So here is my question

In the solution, it said the generating function, $g(x)$ = $(1+ {1 over x} + {1 over x^2})f(x)$

But, In my thought

$(1+ {1 over x} + {1 over x^2})f(x) = S_6x^5 + S_6 x^4 + S_7x^5 + sum_{n=6} ^infty (S_n + S_{n+1} + S_{n+2})x^n = S_6x^5 + S_6 x^4 + S_7x^5 + g(x)$

(I.e. $g(x) = (1+ {1 over x} + {1 over x^2})f(x) – (S_6x^5 + S_6 x^4 + S_7x^5) $ with $S_6 = 1$ and $S_7 =1$)

The reason why $S_6 =1$ is there is a only root $(1,1,1)$. Vice versa $S_7 =1$, having a only root $(2,1,1)$

So my guess is the solution is false. Is my thought right? Any help or advice would be appreciated. Thanks.