I want to find a general formula for calculating:

$$\sum\_{i=1}^n i F\_i $$

where $F\_n$ is the n-th Fibonacci number.
What I have done so far is:
$$\sum\_{i=1}^n \sum\_{j=i+1}^n F\_j$$

I tried to solve it using above double summation formula but it doesn't give the correct result

$F\_1$ = 1,
$F\_2$ = 2,
$F\_3$ = 3,
$F\_4$ = 5.

(for e.g. summation upto 12 is 3916, upto 5 is 46)