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  4. On the Universal Regularity of the Numbers of Generalized Recurrence Sequence and Solutions to Its Characteristic Equation of Second Order

On the Universal Regularity of the Numbers of Generalized Recurrence Sequence and Solutions to Its Characteristic Equation of Second Order

CDS.
2019;
Volume 1, Number 1
: pp. 27 - 33
https://doi.org/10.23939/cds2019.01.027
Authors:
  1. P. Kosobutskyi
1
Lviv Polytechnic National University, Ukraine

In this work shows that the classical oscillations of the ratio of neighboring members of the Fibonacci sequences are valid for arbitrary directions on the plane of the phase coordinates, approaching, to a maximum, the solutions to the characteristic quadratic equation at a given point. The values of the solutions to the characteristic equation along the satellites are asymptotically close to their integer values of the corresponding root lines.

Golden ratio
Phidias number
the quadratic equation
second order recursive sequence
  1. Kosobutskyy P. Modelling of electrodynamic Systems by the Method of Binary Seperation of Additive Parameter in Golden Proportion. Jour. of Electronic Research and Application, 2019,3(3), р. 8–12,
  2. Kosobutskyy P. et.al. Physical principles of Optimization of the Static Regime of a Cantilever-Type Power- effect Sensor with a Constant Rectangular Cross Section. Jour. of Electronic Research and Application, 2018, 2(5), р. 11–15. https://doi.org/10.26689/jera.v2i5.585
  3. Vorobyov N. Fibonacci Numbers. Moscow,1961.
  4. R. Dunlap. The golden ratio and Fibonacci numbers. World Scientific Publishing Co. Pte. Ltd. 1997
  5. Vajda S. (1989) Fibonacci & Lucas Numbers, and the Golden Section. Theory and Applications. Ellis Horwood limited.
  6. Koshy T. (2001) Fibonacci and Lucas numbers with application, A Wiley-Interscience Publication: New York. 
  7. Horadam A. Basic Properties of a Certain Generalized Sequence of Numbers. Fibonacci Quarterly, 3.3(1965), рр. 161–176.
  8. Larcombe P. Horadam Sequences: A Survey Update and Extension , Bulletin of the ICA, Vol. 80 (2017), 99–118. 
  9. F. Gatta,   A. D’amico. Sequences {Hn} for which Hn+1/Hn approaches the Golden Ratio. Fibonacci Quarterly, 46/47.4 (2008/2009), рр. 346–349.
  10. Ozvatan        M.,      Pashev   O.     Generalized          Fibonacci         Sequences          and      Binnet-Fibonacci             Curves. arXiv:1707.09151v1 [math.HO] 28 Jul 2017. https://arxiv.org/pdf/1707.09151.pdf
  11. Szakacs T. K-order Linear Recursive Sequences and the Golden Ratio. Fibonacci Quarterly, 55.5 (2017), рр. 186–191.
  12. Shneider R. Fibonacci numbers and the golden ratio. VarXiv:1611.07384v1 [math.HO] 22 Nov 2016.