Generalized Fibonacci matrices

A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 15, 2009, Number 1, Pages 12–21
Full paper (PDF, 227 Kb)


Authors and affiliations

A. G. Shannon
Warrane College, University of New South Wales
PO Box 123, Kensington, NSW 1465, Australia


Medical measurements, by their very nature, are array-oriented. Matrices are, in a sense, their natural medium of display. Fibonacci matrices, given the natural growth modeling of second order linear sequences, are then particularly suitable vehicles for displaying, developing and discussing medical phenomena. This paper illustrates some of these aspects, partly for their pure mathematical elegance, and partly for their applied mathematical aptness.


  • Fibonacci numbers
  • Recurrence relations
  • False positives
  • Diabetes mellitus
  • Breast cancer
  • Mammography
  • Ultrasonography

AMS Classification

  • 92B05
  • 11B39
  • 11C20


  1. Brousseau, Br Alfred. 1965. Seeking the Lost Gold Mine or Exploring for Fibonacci Factorization. The Fibonacci Quarterly. 3: 129-130.
  2. Chiarella, C., A.G. Shannon. 1986. An Example of Diabetes Compartment Modelling. Mathematical Modelling. 7: 1239-1244.
  3. Geraghty, D.P., A.G. Shannon, S. Colagiuri. 1986. Polynomial Curve-fitting of Clinical Data Arrays. Journal of Clinical Computing. 15(1): 29-32.
  4. Harvey, J.A. 2007. Unusual Breast Cancers: Useful Clues to Expanding the Differential Diagnosis. Radiology. 242: 683-694.
  5. Henderson, J.C. 1992. Breast Cancer Therapy – the Price of Success. The New England Journal of Medicine. 326: 1774-1775.
  6. Heyde, C.C. 1981. On Fibonacci (or Lagged Bienaymé-Galton-Watson) Branching Processes. Journal of Applied Probability. 18:583-591.
  7. Horadam, A.F., P. Filipponi. 1991. Cholesky Algorithm Matrices of Fibonacci Type and Properties of Generalized Sequences. The Fibonacci Quarterly. 19: 164-173.
  8. Horadam, A.F., A.G. Shannon. 1988. Asveld’s Polynomials p (n) j . In A.N. Philippou, A.F. Horadam, G.E. Bergum (eds). Applications of Fibonacci Numbers. Volume 2. Dordrecht: Kluwer, pp.163-176.
  9. Hung, W.T., A.G. Shannon, B.S. Thornton. 1994. The Use of a Second-order Recurrence Relation in the Diagnosis of Breast Cancer. The Fibonacci Quarterly. 32: 253-259.
  10. Irving, J., N. Mullineux. 1964. Mathematics in Physics and Engineering. New York: Academic Press, pp.270-279.
  11. Leslie, P.H. 1945. On the Use of Matrices in Certain Population Mathematics, Biometrika. 33 (1945): 183-212.
  12. Love, T.J. 1980. Thermography as an Indicator of Blood Perfusion. Annals of the New York Academy of Sciences. 429-430.
  13. Makhmudov, A. 1983. On Fibonacci’s Model of Infectious Disease. In G.I. Marchuk, L.N. Belykh (eds). Mathematical Modelling in Immunology and Medicine. Amsterdam: North Holland, pp.319-323.
  14. Moghaddamfar, A.R., S. Navid Salehy, S. Nima Salehy. 2008. Certain Matrices Related to the Fibonacci Sequence Having Recursive Entries. Electronic Journal of Linear Algebra. 17: 543-576.
  15. Shannon, A.G., Leon Bernstein. 1973. The Jacobi-Perron Algorithm and the Algebra of Recursive Sequences. Bulletin of the Australian Mathematical Society. 8: 261–277.
  16. Shannon, A.G. J.H. Clarke, L.J. Hills. 1987. Contingency Relations for Infectious Diseases. Computing Mathematical Applications. 14: 829-833.
  17. Shannon, A.G., R.L. Ollerton, D.R. Owens. 1993. A Cholesky Decomposition in Matching Insulin Profiles. In G.E. Bergum, A.N. Philippou, A.F. Horadam (eds). Applications of Fibonacci Numbers. Volume 5. Dordrecht: Kluwer, pp.497-506.
  18. Stevenson, R.W., T.I. Tsakok, J.A. Parsons. 1980. Matched Glucose Responses to Insulin Administered Subcutaneously and Intravenously. Diabetologia. 18: 423-426.
  19. Thornton, B.S., W.T. Hung, C. Hirst. 1992. Diagnostic Model for Local Temporal Thermal Change at the Skin of the Breast during Extended Application of Diagnostic Ultrasound. IMA Journal of Mathematics Applied in Medicine and Biology. 9: 161-175.
  20. Tortora, G.J., N.P. Anagnostakos. 1987. Principles of Anatomy and Physiology, 5th Editionл New York: Harper and Row, p.390.
  21. Wilkinson, J.H. 1965. The Algebraic Eigenvalue Problem. Oxford: Clarendon Press.

Related papers

Cite this paper

Shannon A. G. (2009). Generalized Fibonacci matrices. Notes on Number Theory and Discrete Mathematics, 15(1), 12-21.

Comments are closed.