Aitken sequences and 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) ...

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.