Higher-order Fibonacci numbers

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 numbershFn are directly related to the higher order Hosoya-typeZ numbers. We investigate the limitsFn/Fn−1 and the corresponding equations, the roots of which allow one to write a general expression forhFn. We also report on thehF counting polynomials that give the partition of thehF numbers in contributions arising fromk pairs of disjoint paths of lengthh. It is interesting to see that the partitions ofhF 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.

[1]  H. Hosoya.  Topological Index. A Newly Proposed Quantity Characterizing the Topological Nature of Structural Isomers of Saturated Hydrocarbons , 1971 .

[2]  P. Gennes.  Scaling Concepts in Polymer Physics , 1979 .

[3]  D. Klein,et al.  Transfer‐Matrix Method for Subgraph Enumeration: Applications to Polypyrene Fusenes , 1986 .

[4]  Alexandru T. Balaban,et al.  Graph theory and theoretical chemistry , 1985 .

[5]  Peter Zinn,et al.  List Operations on Chemical Graphs, 6. Comparative Study of Combinatorial Topological Indexes of the Hosoya Type , 1995, J. Chem. Inf. Comput. Sci..

[6]  L B Kier,et al.  Molecular connectivity VII: specific treatment of heteroatoms. , 1976, Journal of pharmaceutical sciences.

[7]  D. Klein,et al.  Fibonacci numbers in the topological theory of benzenoid hydrocarbons and related graphs , 1989 .

[8]  Douglas J. Klein,et al.  Analytical approach to very large benzenoid polymers , 1995, J. Comput. Chem..

[9]  Haruo Hosoya,et al.  Topological Index and Thermodynamic Properties. I. Empirical Rules on the Boiling Point of Saturated Hydrocarbons , 1972 .

[10]  Haruo Hosoya,et al.  Operator technique for obtaining the recursion formulas of characteristic and matching polynomials as applied to polyhex graphs , 1983 .

[11]  H. Hosoya.  Factorization and recursion relations of the matching and characteristic polynomials of periodic polymer networks , 1991 .

[12]  M. Randic.  Characterization of molecular branching , 1975 .

[13]  W. H. T. Davison,et al.  Theory of Resonance Topology of Fully Aromatic Hydrocarbons. I , 1952 .

[14]  R. Stanley.  What Is Enumerative Combinatorics , 1986 .

[15]  L. Hall,et al.  Molecular connectivity in chemistry and drug research , 1976 .

[16]  C Safran.  The Fibonacci numbers. , 1991, M.D. computing : computers in medical practice.

[17]  A. F. Horadam,et al.  A Generalized Fibonacci Sequence , 1961 .