The Pascal–Fibonacci numbers

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 19, 2013, Number 3, Pages 5—11
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Authors and affiliations

J. V. Leyendekkers
Faculty of Science, The University of Sydney
NSW 2006, Australia

A. G. Shannon
Faculty of Engineering & IT, University of Technology Sydney
NSW 2007, Australia

Abstract

The Pascal–Fibonacci (PF) numbers for a given Fibonacci number sum to give the values of that Fibonacci number. Individual PF numbers are members of one of the triangular, tetrahedral or pentagonal series or have factors in common with the pyramidal or other geometric series. For composite numbers, partial sums of PF numbers can yield a factor, while prime Fibonacci numbers are detected with sums of squares.

Keywords

  • Fibonacci numbers
  • Binet equation
  • Pascal Triangle
  • Triangular numbers
  • Tetrahedral numbers
  • Pentagonal numbers
  • Pyramidal numbers

AMS Classification

  • 11B39
  • 11B50

References

  1. Leyendekkers, J. V., A. G. Shannon. Fibonacci Numbers within Modular Rings. Notes on Number Theory and Discrete Mathematics. Vol. 4, 1998, No. 4, 165–174.
  2. Leyendekkers, J. V., A. G. Shannon. The Structure of Geometric Number Sequences. Notes on Number Theory and Discrete Mathematics. Vol. 17, 2011, No. 3, 31–37.
  3. Leyendekkers, J. V., A. G. Shannon. Geometric and Pellian Sequences. Advanced Studies in Contemporary Mathematics. Vol. 22, 2012, No. 4, 507-508.
  4. Leyendekkers, J. V., A. G. Shannon. The Structure of the Fibonacci Numbers in the Modular Ring Z5. Notes on Number Theory and Discrete Mathematics. Vol. 19, 2013, No. 1, 66–72.
  5. Leyendekkers, J. V., A. G. Shannon. On the Golden Ratio. Advanced Studies in Contemporary Mathematics. Vol. 23, 2013, No. 1, 195–201.
  6. Leyendekkers, J. V., A. G. Shannon. Fibonacci and Lucas Primes. Notes on Number Theory and Discrete Mathematics. Vol. 19, 2013, No. 2, 49–59.
  7. Shannon, A. G. Tribonacci Numbers and Pascal’s Pyramid. The Fibonacci Quarterly. Vol. 15, 1977, No. 3, 268, 275.
  8. Wong, C. K., T. W. Maddocks. A Generalized Pascal’s Triangle. The Fibonacci Quarterly. Vol. 13, 1975, No. 2, 134–136.

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Cite this paper

Leyendekkers, J. V., & Shannon, A. (2013). The Pascal–Fibonacci numbers. Notes on Number Theory and Discrete Mathematics, 19(3), 5-11.

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