We investigate several methods of computing Fibonacci numbers quickly and generalize some properties of the Fibonacci numbers to degree r Fibonacci (R-nacci) numbers. Sections 2 and 3 present several algorithms for computing the traditional, degree two, Fibonacci numbers quickly. Sections 4 and 5 investigate the structure of the binary representation of the Fibonacci numbers. Section 6 shows how the generalized Fibonacci numbers can be expressed as rounded powers of the dominant root of the characteristic equation. Properties of the roots of the characteristic equation of the generalized Fibonacci numbers are presented in Section 7. Section 8 introduces several properties of the Zeckendorf representation of the integers. Finally, in Section 9 the asymptotic proportion of l’s in the Zeckendorf representation of integers is computed and an easy to compute closed formula is given.
KeywordsBinary Sequence Compression Algorithm Binary Representation Fibonacci Number Fibonacci Sequence
Unable to display preview. Download preview PDF.
- Capocelli, R. M., “A generalization of Fibonacci Trees,” Third International Conference on Fibonacci Numbers and Their Applications, Pisa, Italy; 1988.Google Scholar
- Capocelli, R. M., and P. Cull, “Generalized Fibonacci numbers are rounded powers,” Third International Conference on Fibonacci Numbers and Their Applications, Pisa, Italy; 1988.Google Scholar
- Feinberg, M., “Fibonacci-Tribonacci,” The Fibonacci Quarterly 1, 71–74; 1963.Google Scholar
- Gallager, R. G., “Vaxiations on a theme of Huffman,” IEEE Transactions on Information Theory IT-24, 668–674; 1978.Google Scholar
- Knuth, D. E., “The Art of Computer Programming,” Vols. 1 and 2, Addison- Wesley, Reading MA; 1973, 1981.Google Scholar
- Romano, A., “Applied statistics for science and industry,” Allyn and Bacon, Boston; 1977.Google Scholar
- Vorob’ev, N.N., “Fibonacci Numbers,” Pergamon Press, New York; 1961.Google Scholar
- Zeckendorf, E., “Representations des Nombres Naturels par une Somme de Nombres de Fibonacci ou de Nombres de Lucas,” Bulletin de la Société Royale des Sciences de Liege 179–182; 1972.Google Scholar