On the self-convolution of generalized Fibonacci numbers

ON r-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.

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 (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.

On the Growth Rate of Generalized Fibonacci Numbers

for all nonnegative integers i, j such that j ≤ i, as illustrated in Figure 1.1. The points in R2 associated with ( i j ) , ( i+1 j ) , and ( i+1 j+1 ) form a unit equilateral triangle. This arrayal is called the natural arrayal of Pascal’s triangle in R2. For all t ∈ R : −√3 < t < √3 and nonnegative integers k, define k(t) to be the sum of all binomial coefficients associated with points in R2...