Advertisement

On Matrix Representations of Generalized Fibonacci Numbers and Their Applications

  • Shuichi Sato

Abstract

Many kinds of generalizations of Fibonacci numbers (e.g., see [2], [4], [7], [13], [15]) have been investigated and many interesting properties of the generalized Fibonacci numbers have been obtained. Furthermore, some notable generalizations of the Lucas Number (e.g., see [3], [6], [8]) have been undertaken.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Bergum, G.E., Bennett, L., Horadam, A.F. and Moore, S.D. “Jacobsthal Polynomials and a Conjecture Concerning Fibonacci-like Matrices.” The Fibonacci Quarterly, Vol. 23.3 (1985): pp. 240–248.zbMATHGoogle Scholar
  2. [2]
    Bergum, G.E. and Hoggatt, V.E. Jr. “An Application of the Characteristic of the Generalized Fibonacci Sequence.” The Fibonacci Quarterly, Vol. 15.3 (1977): pp. 215–220.MathSciNetzbMATHGoogle Scholar
  3. [3]
    Clarke, J.H. and Shannon, A.G. “Some Generalized Lucas Sequences.” The Fibonacci Quarterly, Vol. 23.2 (1985): pp. 120–125.MathSciNetzbMATHGoogle Scholar
  4. [4]
    Feinberg, Mark. “Fibonacci-Tribonacci.” The Fibonacci Quarterly, Vol. 1.3 (1963): pp. 71–74.Google Scholar
  5. [5]
    Filipponi, P. and Horadam, A.F. “A Matrix Approach to Certain Identities.” The Fibonacci Quarterly, Vol. 26.2 (1988): pp. 115–126.MathSciNetzbMATHGoogle Scholar
  6. [6]
    Hoggatt, V.E. Jr. and Bicknell-Johnson, M. “Generalized Lucas Sequences.” The Fibonacci Quarterly, Vol. 15.2 (1977): pp. 131–139.MathSciNetzbMATHGoogle Scholar
  7. [7]
    Horadam, A.F. “A Generalized Fibonacci Sequence.” The American Mathematical Monthly, Vol. 68.5 (1961): pp. 455–459.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Horadam, A.F. “Generating Identities for Generalized Fibonacci and Lucas Triples.” The Fibonacci Quarterly, Vol. 15.4 (1977): pp. 289–292.MathSciNetzbMATHGoogle Scholar
  9. [9]
    Horadam, A.F. and Filipponi, P. “Cholesky Algorithm Matrices of Fibonacci Type and Properties of Generalized Sequences.” The Fibonacci Quarterly, Vol. 29.2 (1991): pp. 164–173.MathSciNetzbMATHGoogle Scholar
  10. [10]
    Liu, Bolian. “A Matrix Method to Solve Linear Recurrences with Constant Coefficients.” The Fibonacci Quarterly, Vol. 30.1 (1992): pp. 2–8.MathSciNetzbMATHGoogle Scholar
  11. [11]
    Mahon, Bro.J.M. and Horadam, A.F. “Matrix and Other Summation Techniques for Pell Polynomials.” The Fibonacci Quarterly, Vol. 24.4 (1986): pp. 290–308.MathSciNetzbMATHGoogle Scholar
  12. [12]
    Miles, E.P. Jr. “Generalized Fibonacci Numbers and Associated Matrices.” The American Mathematical Monthly, Vol. 67.10 (1960): pp. 745–752.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    Miller, M.D. “On Generalized Fibonacci Numbers.” The American Mathematical Monthly, Vol. 78.10 (1971): pp. 1108–1109.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    Vorob’ev, N.N. Fibonacci Numbers. Pergamon Press (translated from the Russian) (1961).Google Scholar
  15. [15]
    Waddill, M.E. “The Tetranacci Sequence and Generalizations.” The Fibonacci Quarterly, Vol. 30.1 (1992): pp. 9–20.MathSciNetzbMATHGoogle Scholar
  16. [16]
    Waddill, M.E. and Sacks, Louis. “Another Generalized Fibonacci Sequence.” The Fibonacci Quarterly, Vol. 5.3 (1967): pp. 209–222.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1993

Authors and Affiliations

  • Shuichi Sato

There are no affiliations available

Personalised recommendations