Some variations on Fibonacci matrix graphs

Anthony G. Shannon and Ömür Deveci
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 23, 2017, Number 4, Pages 85–93
Full paper (PDF, 201 Kb)

Details

Authors and affiliations

Anthony G. Shannon
Emeritus Professor, University of Technology Sydney, NSW 2007,
Fellow, Warrane College, University of New South Wales, Kensington NSW 2033,
Director, Academic Affairs, Australian Institute of Music, Sydney NSW 2010, Australia

Ömür Deveci
Department of Mathematics, Faculty of Science and Letters,
Kafkas University 36100, Turkey

Abstract

Matrices are here considered in two ways: arrays containing Fibonacci numbers and their generalizations in the cells, and arrays as graphs where the cells themselves are subgraphs. Both aspects contain ideas for further development and research.

Keywords

• Fibonacci
• Pell and Eulerian numbers
• Pyramidal numbers
• Golden section
• Spanning trees
• Lattice points

AMS Classification

• 11B39
• 05C62

References

1. Benson, D. (2006) Music: A Mathematical Offering, Cambridge University Press, Cambridge.
2. Cahill, N., & Narayan, D. (2004) Fibonacci and Lucas numbers in tridiagonal matrix determinants, The Fibonacci Quarterly, 42(3), 216–221.
3. Deveci, O., & Shannon, A. G. (2017) On the adjacency-type sequences, International Journal of Advances in Mathematics, 17(2), 10–24.
4. Hilton, A. J. W. (1974) Spanning trees and Fibonacci and Lucas numbers, The Fibonacci Quarterly, 12(3), 259–262.
5. Horadam, A. F. (1965) Basic properties of a certain generalized sequence of numbers, The Fibonacci Quarterly, 3(3), 161–176.
6. Horadam, A. F., & Mahon, J. M. (1986) Convolutions for Pell polynomials. In A.N. Philippou, Gerald E. Bergum and Alwyn F. Horadam (eds). Fibonacci Numbers and Their Applications, Reidel, Dordecht, 55–80.
7. Jarden, D. (1958) Recurring Sequences. Riveon Lematematika, Jerusalem, 1–4.
8. Kuijken, B. (2013) The Notation is Not the Music, Indiana University Press, Bloomington & Indianapolis, IN.
9. Ollerton, R. L., & Shannon, A. G. (1998) Some properties of generalized Pascal squares and triangles, The Fibonacci Quarterly, 36(2), 98–109.
10. Pruitt, R. (1967) Fibonacci and Lucas numbers in the sequence of golden numbers, The Fibonacci Quarterly, 5(2), 175–178.
11. Quaintance, J., & Gould, H. W. (2015) Combinatorial Numbers for Stirling Numbers: The Unpublished Notes of H. W. Gould, World Scientific, New Jersey, Ch. 10.
12. Rebman, K. R. (1975) The sequence 1 5 16 45 121 320 … in Combinatorics, The Fibonacci Quarterly, 13(1), 51–55.
13. Shannon, A. G. (1974) Some properties of a fundamental recursive sequence of arbitrary order, The Fibonacci Quarterly, 12(4), 327–335.
14. Shannon, A. G. (1978), Fibonacci and Lucas Numbers and the Complexity of a Graph, The Fibonacci Quarterly, 16(1), 1–4.
15. Sloane, N. J. A., & Plouffe, S. (1968) The Encyclopedia of Integer Sequences, Academic Press, San Diego, CA.
16. Stein, S. K. (1962) The intersection of Fibonacci sequences, Michigan Mathematical Journal, 9(4), 399–402.
17. Whitford, A. K. (1977) Binet’s formula generalized, The Fibonacci Quarterly, 15(1), 21, 24, 29.