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On Sums of Cubes of Fibonacci Numbers

  • Chapter
Applications of Fibonacci Numbers

Abstract

Our story begins with one of the simplest, prettiest, and easiest to prove of all the Fibonacci summation formulas, the formula

$$ \sum\limits_{{k = 1}}^n {F_k^2 = {F_n}{F_{{n + 1}}}} $$
(1)

for the sum of the squares of the consecutive Fibonacci numbers. We were struck by the elegance of this formula—especially by its expressing the sum in factored form—and wondered whether anything similar could be done for sums of cubes of Fibonacci numbers. This paper is a report of some of our discoveries.

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References

  1. Allison, D. “A note on sums of powers of integers.” The American Mathematical Monthly, Vol. 68.3 (1961): p. 272.

    Article  MathSciNet  Google Scholar 

  2. Block, D. “Curiosum #330: Fibonacci summations.” Scripta Mathematica, Vol. 19.2–3 (1953): p. 191.

    Google Scholar 

  3. Brousseau, Brother A. Fibonacci and Related Number Theoretic Tables. The Fibonacci Association, 1972.

    Google Scholar 

  4. Cavior, S.R. “A theorem on power sums.” The Fibonacci Quarterly, Vol. 6.2 (1968): pp. 157–161.

    MathSciNet  MATH  Google Scholar 

  5. Edmonds, S.M. “Sums of powers of the natural numbers.” The Mathematical Gazette, Vol. 41.337 (1957): pp. 187–188.

    Article  MathSciNet  MATH  Google Scholar 

  6. Euler, L. “Summatio progressionum sin.ø⋋ + sin.2ø⋋ + sin.3ø⋋ + … + sin.nø⋋, cos.ø⋋ + cos.2ø⋋ + cos.3ø⋋ + … + cos.nø⋋.” Novi commentarii academiae scientiarum Petropolitanae, Vol. 18 (1773): pp. 24–36. Reprinted in Euler’s Opera Omnia, series 1, volume 15, pp. 168–184.

    Google Scholar 

  7. Horadam, A.F. “Basic properties of a certain generalized sequence of numbers.” The Fibonacci Quarterly, Vol. 3.3 (1965): pp. 161–176.

    MathSciNet  MATH  Google Scholar 

  8. Klamkin, M.S. “On some identities of Lucas.” Scripta Mathematica, Vol. 21.2–3 (1955): pp. 213-214.

    MATH  Google Scholar 

  9. Krishna, H.V. “Identities of a generalized Fibonacci sequence.” In Verner E. Hoggatt, Jr. and Marjorie Bicknell-Johnson, editors, A Collection of Manuscripts Related to the Fibonacci Sequence: 18th Anniversary Volume, pp. 65–66. The Fibonacci Association, Santa Clara, California, 1980.

    Google Scholar 

  10. Lucas, E. “Théorie des fonctions numériques simplement périodiques.” American Journal of Mathematics, Vol. 1 (1878): pp. 184–240. English translation of the first part in [11].

    Article  MathSciNet  Google Scholar 

  11. Lucas, E. “Théorie des fonctions numériques simplement périodiques.” American Journal of Mathematics, Vol. 1 (1878): pp. 289–321.

    Article  MathSciNet  Google Scholar 

  12. Lucas, E. The Theory of Simply Periodic Numerical Functions. The Fibonacci Association, 1969. A translation of the first part of [10] by Sidney Kravitz, edited by Douglas Lind.

    Google Scholar 

  13. Lucas, E. Théorie des Nombres. Volume I. Gauthier-Villars et Fils, Paris, 1891.

    MATH  Google Scholar 

  14. Mahon, Brother J.M. and Horadam, A.F. “Matrix and other summation techniques for Pell polynomials.” The Fibonacci Quarterly, Vol. 24.4 (1986): pp. 290–309.

    MathSciNet  MATH  Google Scholar 

  15. National Bureau of Standards. Tables of Chebyshev Polynomials S n(x) and Cn(x). United States Government Printing Office, December 19, 1952. Applied Mathematics Series, Number 9. Introduction by Cornelius Lanczos.

    Google Scholar 

  16. Penney, D.E. and Pomerance, C. “Multiplicative relations for sums of initial kth. powers.” The American Mathematical Monthly, Vol. 92.10 (1985): pp. 729–731.

    Article  MathSciNet  MATH  Google Scholar 

  17. Pond, J.C. “Generalized Fibonacci summations.” The Fibonacci Quarterly, Vol. 6.2 (1968): pp. 97–108.

    MathSciNet  MATH  Google Scholar 

  18. Rivlin, T.J. Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory. John Wiley & Sons, second edition, 1990.

    MATH  Google Scholar 

  19. Subba Rao, K. “Some properties of Fibonacci numbers.” The American Mathematical Monthly, Vol. 60.10 (1953): pp. 680–684.

    Article  MathSciNet  MATH  Google Scholar 

  20. Utz, W.R. “The Diophantine equation (x 1 + x 2 + + x n)2 = x 3 1 + x 3 n.” The Fibonacci Quarterly, Vol. 15.1 (1977): pp. 14

    MathSciNet  Google Scholar 

  21. Utz, W.R. “The Diophantine equation (x 1 + x 2 + + x n)2 = x 3 1 + x 3 n.” The Fibonacci Quarterly, Vol. 15.1 (1977): pp. 16.

    MathSciNet  Google Scholar 

  22. Vorob’ev, N.N. Fibonacci Numbers. Blaisdell, New York-London, 1961.

    MATH  Google Scholar 

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Authors
  1. Stuart Clary
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  2. Paul D. Hemenway
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Editor information

Editors and Affiliations

  1. Computer Science Department, South Dakota State University, Box 2201, Brookings, South Dakota, 57007-0194, USA

    Gerald E. Bergum

  2. Ministry of Education, Government House Z 50, Nicosia, Cyprus

    Andreas N. Philippou

  3. Department of Mathematics, Statistics and Computer Science, University of New England, Armidale, New South Wales, 2351, Australia

    Alwyn F. Horadam

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© 1993 Springer Science+Business Media Dordrecht

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Clary, S., Hemenway, P.D. (1993). On Sums of Cubes of Fibonacci Numbers. In: Bergum, G.E., Philippou, A.N., Horadam, A.F. (eds) Applications of Fibonacci Numbers. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2058-6_12

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  • DOI: https://doi.org/10.1007/978-94-011-2058-6_12

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-4912-2

  • Online ISBN: 978-94-011-2058-6

  • eBook Packages: Springer Book Archive

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