Let Fibonacci numbers be defined as usual, represented by homogeneous linear recurrence $F_n = F_{n-1} + F_{n-2}$, with starting coefficients $F_0 = 0, F_1 = 1$. Consider sequence $G_n = n\cdot{F_n}$, defined for non-negative integers $n$. How do I represent $G_n$ by a homogeneous linear reccurence?
Computing $G_n$ gives me a following sequence (A045925):
$$ 0, 1, 2, 6, 12, 25, 48, 91, 168, 306, 550,\cdots$$
I have been going at this question for a while, and have searched all over this website too, haven't been able to find anything. I gather that there is some relationship between general solutions and characteristic equations of $F_n$ and $G_n$, and I intuitively suspect general form to be $$nF(n) = (c_1+c_2n)(\varphi)^n+(c_3+c_4n)(1-\varphi)^n$$ But apart from that this question escapes me.
The generating function for the Fibonacci numbers has denominator $1-x-x^2$. Multiplying by $n$ should replace the denominator by its square, viz., $$(1-x-x^2)^2=1-2x-x^2+2x^3+x^4.$$ Therefore I'd expect $nF_n$ to satisfy the recurrence $$a_n=2a_{n-1}+a_{n-2}-2a_{n-3}-a_{n-4}.$$