What is the solution to $F\left(n\right)=F\left(n-1\right)+\left(\frac{2}{n}\right)F\left(n-2\right)$ with Legendre symbol?

Question

The recurrence $F\left(n\right)=F\left(n-1\right)+\left(\frac{2}{n}\right)F\left(n-2\right) $ was proposed here

Because of the way the question was formatted, a commenter asked if the $\left(\frac{2}{n}\right) $ meant the Legendre symbol. The proposer said no, it was just ordinary division.

That made me wonder, suppose the Legendre symbol was meant. What would the solution be?

So that is my question.

I have no idea how to solve it.