A monotonicity property for generalized Fibonacci sequences

On Periodic ∞-generalized Fibonacci Sequences

The notion of an∞-generalized Fibonacci sequence has been introduced in [6], and studied in [1], [7], [9]. This class of sequences defined by linear recurrences of infinite order is an extension of the class of ordinary (weighted) r-generalized Fibonacci sequences (r-GFS, for short) with r finite defined by linear recurrences of r order (for example, see [2], [3], [4], [5], [8] etc.) More preci...

p-ADIC CONGRUENCES FOR GENERALIZED FIBONACCI SEQUENCES

is the ordinary formal power series generating function for the sequence {yn+i}„>0 (cf. [12]. Furthermore, it is easy to see [1] that when the discriminant A = X +4ju ofP(t) is nonnegative and X & 0, the ratios yn+l I yn converge (in the usual archimedean metric on U) to a reciprocal root a of P(t). In this article we show that ratios of these y n also exhibit rapid convergence properties relat...

Some Divisibility Properties of Generalized Fibonacci Sequences

Let c be any square-free integer, p any odd prime such that (c/p) = -1, and n any positive integer. The quantity ./IT, which would ordinarily be defined (mod p) as one of the two solutions of the congruence: x E c (mod p n ) , does not exist. Nevertheless, we may deal with objects of the form a + b/c~(mod p), where a and b are integers, in much the same way that we deal with complex numbers, th...

Waiting times and Generalized Fibonacci Sequences

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) ...