Abstract
Our story begins with one of the simplest, prettiest, and easiest to prove of all the Fibonacci summation formulas, the formula
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.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Allison, D. “A note on sums of powers of integers.” The American Mathematical Monthly, Vol. 68.3 (1961): p. 272.
Block, D. “Curiosum #330: Fibonacci summations.” Scripta Mathematica, Vol. 19.2–3 (1953): p. 191.
Brousseau, Brother A. Fibonacci and Related Number Theoretic Tables. The Fibonacci Association, 1972.
Cavior, S.R. “A theorem on power sums.” The Fibonacci Quarterly, Vol. 6.2 (1968): pp. 157–161.
Edmonds, S.M. “Sums of powers of the natural numbers.” The Mathematical Gazette, Vol. 41.337 (1957): pp. 187–188.
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.
Horadam, A.F. “Basic properties of a certain generalized sequence of numbers.” The Fibonacci Quarterly, Vol. 3.3 (1965): pp. 161–176.
Klamkin, M.S. “On some identities of Lucas.” Scripta Mathematica, Vol. 21.2–3 (1955): pp. 213-214.
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.
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].
Lucas, E. “Théorie des fonctions numériques simplement périodiques.” American Journal of Mathematics, Vol. 1 (1878): pp. 289–321.
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.
Lucas, E. Théorie des Nombres. Volume I. Gauthier-Villars et Fils, Paris, 1891.
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.
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.
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.
Pond, J.C. “Generalized Fibonacci summations.” The Fibonacci Quarterly, Vol. 6.2 (1968): pp. 97–108.
Rivlin, T.J. Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory. John Wiley & Sons, second edition, 1990.
Subba Rao, K. “Some properties of Fibonacci numbers.” The American Mathematical Monthly, Vol. 60.10 (1953): pp. 680–684.
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
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.
Vorob’ev, N.N. Fibonacci Numbers. Blaisdell, New York-London, 1961.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
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
Download citation
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