Unit coefficient sums for certain Morgan–Voyce numbers

A. F. Horadam
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 3, 1997, Number 3, Pages 117—127
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A. F. Horadam
University of New England, Armidale, Australia 2351

Abstract

Studying the unique minimal and maximal integer Zeckendorf representations by Pell numbers [2], [3], [4], [5] led to the consideration [6, eqn. (2.7)] of those numbers which are common to both representations, namely, the MinMax numbers. This idea was then carried over to Jacobsthal numbers [7, eqn. (3.1)]. (Earlier, minimal and maximal representations by Fibonacci and Lucas numbers had been investigated in [1].)
Here, the corresponding situation existing for Morgan–Voyce numbers is to be disclosed. Though the results are perhaps not quite so elegant as those for Pell numbers, they are nevertheless of intrinsic interest and value.

AMS Classification

  • 11B37

References

  1. V.E. Hoggatt, Jr. Fibonacci and Lucas Numbers. Houghton Mifflin (1969).
  2. A.F. Horadam. “Zeckendorf Representations of Positive and Negative Integers by Pell Numbers.” Applications of Fibonacci Numbers, Vol.5 (ed. G.E. Bergum, A.N. Philippou, and A.F. Horadam), Kluwer Academic Publications, Dordrecht (1993): 305-316.
  3. A.F. Horadam. “Unique Minimal Representation of Integers by Negatively Subscripted Pell Numbers.” The Fibonacci Quarterly 32.3 (1994) 202-206.
  4. A.F. Horadam. “Maximal Representations of Positive Integers by Pell Numbers.” The Fibonacci Quarterly 32.3 (1994): 240-244.
  5. A.F. Horadam. “An Alternative Proof of a Unique Representation Theorem.” The Fibonacci Quarterly 32.5 (1994): 409-411.
  6. A.F. Horadam. “MinMax Sequences for Pell Numbers.” Applications of Fibonacci Numbers, Vol.6 (ed. G.E. Bergum, A.N. Philippou and A.F. Horadam), Kluwer Academic Publications, Dordrecht (1996): 231-249.
  7. A.F. Horadam. “Jacobsthal Representation Numbers.” The Fibonacci Quarterly 34.1 (1996): 40-54.
  8. A.F. Horadam. “New Aspects of Morgan-Voyce Polynomials.” (To appear in the proceedings of the Seventh International Conference on Fibonacci Numbers and Their Applications.)
  9. A.M. Morgan-Voyce. “Ladder Networks Analysis Using Fibonacci Numbers.” I.R.E. Trans. Circuit Theory 6.3 (1959): 321-322.
  10. N.N. Vorob’ev. Fibonacci Numbers. Pergamon (1961).

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Cite this paper

APA

Horadam, A. F. (1997). Unit coefficient sums for certain Morgan—Voyce numbers. Notes on Number Theory and Discrete Mathematics, 3(3), 117-127.

Chicago

Horadam, A. F. “Unit coefficient sums for certain Morgan—Voyce numbers.” Notes on Number Theory and Discrete Mathematics 3, no. 3 (1997): 117-127.

MLA

Horadam, A. F. “Unit coefficient sums for certain Morgan—Voyce numbers.” Notes on Number Theory and Discrete Mathematics 3.3 (1997): 117-127. Print.

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