Solution
= Solution
\b[Use $G_1=0$ and $G_{n+1}=nG_n+n-1$ for $n\ge1$.] Indeed,
$$
G_{n+1}=n!-1=n\big((n-1)!-1\big)+n-1=nG_n+n-1.
$$
The initial value is $0!-1=0$, so the <recurrence relation> produces the specified <factorial> sequence.
= Solution
\b[Use $G_1=0$ and $G_{n+1}=nG_n+n-1$ for $n\ge1$.] Indeed,
$$
G_{n+1}=n!-1=n\big((n-1)!-1\big)+n-1=nG_n+n-1.
$$
The initial value is $0!-1=0$, so the <recurrence relation> produces the specified <factorial> sequence.