Abstract
We consider a generalization of Fibonacci numbers that was motivated by the relationship of the HosoyaZ topological index to the Fibonacci numbers. In the case of the linear chain structures the new higher order Fibonacci numbersh F n are directly related to the higher order Hosoya-typeZ numbers. We investigate the limitsF n /F n−1 and the corresponding equations, the roots of which allow one to write a general expression forhFn. We also report on theh F counting polynomials that give the partition of theh F numbers in contributions arising fromk pairs of disjoint paths of lengthh. It is interesting to see that the partitions ofh F are “hidden” in the Pascal triangle in a similar way to the partitions of the Fibonacci numbers that were discovered some time ago by Hoggatt. We end with illustrations of the recursion formulas for the higher order Hosoya numbers for several families of graphs that are based on the corresponding recursions for the higher Fibonacci numbers.
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Dedicated to Professor Haruo Hosoya of Ochanomizu University, Tokyo, Japan, on the occasion of 25 years of the topological index Z.
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Randić, M., Morales, D.A. & Araujo, O. Higher-order Fibonacci numbers. J Math Chem 20, 79–94 (1996). https://doi.org/10.1007/BF01165157
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DOI: https://doi.org/10.1007/BF01165157