Generalized Fibonacci Numbers, Cosmological Analogies, and an Invariant

Restricted Permutations, Fibonacci Numbers, and k-generalized Fibonacci Numbers

In 1985 Simion and Schmidt showed that the number of permutations in Sn which avoid 132, 213, and 123 is equal to the Fibonacci number Fn+1. We use generating function and bijective techniques to give other sets of pattern-avoiding permutations which can be enumerated in terms of Fibonacci or k-generalized Fibonacci numbers.

GENERALIZED q - FIBONACCI NUMBERS

We introduce two sets of permutations of {1, 2, . . . , n} whose cardinalities are generalized Fibonacci numbers. Then we introduce the generalized q-Fibonacci polynomials and the generalized q-Fibonacci numbers (of first and second kind) by means of the major index statistic on the introduced sets of permutations.

ON r-GENERALIZED FIBONACCI NUMBERS

Generalized (k, r)–Fibonacci Numbers

In this paper, and from the definition of a distance between numbers by a recurrence relation, new kinds of k–Fibonacci numbers are obtained. But these sequences differ among themselves not only by the value of the natural number k but also according to the value of a new parameter r involved in the definition of this distance. Finally, various properties of these numbers are studied.

An Identity in the Generalized Fibonacci Numbers and Its Applications

We first generalize an identity involving the generalized Fibonacci numbers and then apply it to establish some general identities concerning special sums. We also give a sufficient condition on a generalized Fibonacci sequence {Un} such that Un is divisible by an arbitrary prime r for some 2 < n ≤ r − 2. 1. Preliminaries The generalized Fibonacci and Lucas numbers are defined, respectively, by...