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On the Fibonacci k-numbers

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  • Falcón, Sergio
  • Plaza, Ángel

Abstract

We introduce a general Fibonacci sequence that generalizes, between others, both the classic Fibonacci sequence and the Pell sequence. These general kth Fibonacci numbers {Fk,n}n=0∞ were found by studying the recursive application of two geometrical transformations used in the well-known four-triangle longest-edge (4TLE) partition. Many properties of these numbers are deduce directly from elementary matrix algebra.

Suggested Citation

  • Falcón, Sergio & Plaza, Ángel, 2007. "On the Fibonacci k-numbers," Chaos, Solitons & Fractals, Elsevier, vol. 32(5), pages 1615-1624.
    • Handle: RePEc:eee:chsofr:v:32:y:2007:i:5:p:1615-1624
    DOI: 10.1016/j.chaos.2006.09.022
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    References listed on IDEAS

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    1. Stakhov, Alexey & Rozin, Boris, 2006. "The continuous functions for the Fibonacci and Lucas p-numbers," Chaos, Solitons & Fractals, Elsevier, vol. 28(4), pages 1014-1025.
    2. Stakhov, Alexey & Rozin, Boris, 2005. "The Golden Shofar," Chaos, Solitons & Fractals, Elsevier, vol. 26(3), pages 677-684.
    3. El Naschie, M.S., 2005. "Non-Euclidean spacetime structure and the two-slit experiment," Chaos, Solitons & Fractals, Elsevier, vol. 26(1), pages 1-6.
    4. El Naschie, M.S., 2005. "Dead or alive: Desperately seeking Schrödinger’s cat," Chaos, Solitons & Fractals, Elsevier, vol. 26(3), pages 673-676.
    5. Stakhov, Alexey, 2006. "Fundamentals of a new kind of mathematics based on the Golden Section," Chaos, Solitons & Fractals, Elsevier, vol. 27(5), pages 1124-1146.
    6. El Naschie, M.S., 2005. "Experimental and theoretical arguments for the number and the mass of the Higgs particles," Chaos, Solitons & Fractals, Elsevier, vol. 23(4), pages 1091-1098.
    7. El Naschie, M.S., 2005. "On the cohomology and instantons number in E-infinity Cantorian spacetime," Chaos, Solitons & Fractals, Elsevier, vol. 26(1), pages 13-17.
    8. El Naschie, M.S., 2005. "Anomaly cancellation and the mass spectrum of ε(∞)," Chaos, Solitons & Fractals, Elsevier, vol. 23(3), pages 1087-1090.
    9. Stakhov, Alexey, 2006. "The golden section, secrets of the Egyptian civilization and harmony mathematics," Chaos, Solitons & Fractals, Elsevier, vol. 30(2), pages 490-505.
    10. Naschie, M.S.El, 2005. "Deriving the essential features of the standard model from the general theory of relativity," Chaos, Solitons & Fractals, Elsevier, vol. 24(4), pages 941-946.
    11. Stakhov, A. & Rozin, B., 2006. "The “golden” algebraic equations," Chaos, Solitons & Fractals, Elsevier, vol. 27(5), pages 1415-1421.
    12. El Naschie, M.S., 2005. "Stability Analysis of the two-slit experiment with quantum particles," Chaos, Solitons & Fractals, Elsevier, vol. 26(2), pages 291-294.
    13. Stakhov, A.P., 2005. "The Generalized Principle of the Golden Section and its applications in mathematics, science, and engineering," Chaos, Solitons & Fractals, Elsevier, vol. 26(2), pages 263-289.
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