Golden ratios, Lucas Sequences and the Quadratic Family

Abstract: It is conjectured that there is a converging sequence of some generalized Fibonacci ratios, given the difference between consecutive ratios, such as the Golden Ratio, $\varphi^1$, and the next golden ratio $\varphi^2$. Moreover, the graphic depiction of those ratios show some overlap with the quadratic family, and some numerical evidence suggest that everyone of those ratios in the finite set obtained, belong to at least one quadratic family, and finally a proof is presented that the converging sequence of some generalized Fibonacci ratios belong to at least one quadratic family.