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Golden ratios, Lucas Sequences and the Quadratic Family

Arturo Ortiz Tapia

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.

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, , and the next golden ratio . 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.

Paper Structure

This paper contains 12 sections, 14 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Golden Ratio and Beauty
  3. Lucas Numbers
  4. Other $\varphi$'s
  5. Positive $\varphi$'s
  6. Towards convergence of $\varphi$'s
  7. Matrix form of Lucas Sequences and its eigenvalues
  8. The $\varphi^{k,n}$ and the Quadratic Family
  9. Further Numerical Exploration of Eigenvalues of $A_k,\, k\geq 10$
  10. $\forall \varphi^{k,n}$, $\varphi^{k,n}$ belongs to at least a given $\mathcal{J}$
  11. CONCLUSIONS
  12. Acknowledgements

Figures (9)

  • Figure 1: Plot of all $\varphi^{k,n}$
  • Figure 2: The complete set of $\varphi^{k,n}$ plotted together with $\mathcal{M}$.
  • Figure 3: The complete set of $\varphi^{k,n}$ plotted together with $\mathcal{J}$.
  • Figure 4: Zoom of $\mathcal{J}$ in the region where $\varphi^{1,1}\in \mathcal{J}$.
  • Figure 5: The set of eigenvalues for $A_{10}$ plotted together with $\mathcal{M}$.
  • ...and 4 more figures

Theorems & Definitions (1)

  • proof