Quadratic reciprocity via generalized Fibonacci numbers?

This is a pet idea of mine which I thought I'd share. Fix a prime q congruent to 1mod4 and define a sequence Fn by F0=0,F1=1, andFn+2=Fn+1+q−14Fn.Then Fn=αn−βnα−β where α,β are the two roots of f(x)=x2−x−q−14. When q=5 we recover the ordinary Fibonacci numbers. The discriminant of f(x) is q, so it splits modp if and only if q is a quadratic residue modp. If (qp)=−1, then the Frobenius