Zeckendorf Representation

The Zeckendorf representation of a positive integer n is a representation of n as a sum of nonconsecutive distinct Fibonacci numbers,


where epsilon_k are 0 or 1 and


Every positive integer can be written uniquely in such a form.

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Brown, J. L. Jr. "Zeckendorf's Theorem and Some Applications." Fib. Quart. 2, 163-168, 1964.Keller, T. J. "Generalizations of Zeckendorf's Theorem." Fib. Quart. 10, 95-112, 1972.Lekkerkerker, C. G. "Voorstelling van natuurlijke getallen door een som van Fibonacci." Simon Stevin 29, 190-195, 1951-52.


Fibonacci Cube Graph, Zeckendorf's Theorem


Fraenkel, A. S. "Systems of Numeration." Amer. Math. Monthly 92, 105-114, 1985.Grabner, P. J.; Tichy, R. F.; Nemes, I.; and Pethő, A. "On the Least Significant Digit of Zeckendorf Expansions." Fib. Quart. 34, 147-151, 1996.Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, pp. 295-296, 1994.Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, p. 40, 1991.Zeckendorf, E. "Représentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas." Bull. Soc. Roy. Sci. Liège 41, 179-182, 1972.

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Zeckendorf Representation

Cite this as:

Weisstein, Eric W. "Zeckendorf Representation." From MathWorld--A Wolfram Web Resource.

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