Some Properties of Fibonacci Numbers, Generalized Fibonacci Numbers and Generalized Fibonacci Polynomial Sequences

Some Asymptotic Properties of Generalized Fibonacci Numbers

1. INTRODUCTION Horadam [1] has generalized two theorems of Subba Rao [3] which deal with some asymptotic p r o p e r t i e s of Fibonacci numbers. Horadam defined a sequence {w (n 2) where a , a are the roots of x 2-P 21 x + P 2 2 = 0. We shall let 06 """ LX r\ r\ UO r\-| • Horadam established two theorems for {w n }: I. The number of terms of {w n } not exceeding N is asymptotic to log(Nd/(P ...

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.

Aitken Sequences and Generalized Fibonacci Numbers

Consider the sequence (t>„) generated by t>„+ ¡ = avn bv,l_l, n 5¡ 2, where v¡ = 1, t>2 = a, with a and b real, of which the Fibonacci sequence is a special case. It is shown that if Aitken acceleration is used on the sequence (x„) defined by a„ = v„+1/v„, the resulting sequence is a subsequence of (jc„). Second, if Newton's method and the secant method are used (with suitable starting values) ...

ON r-GENERALIZED FIBONACCI NUMBERS